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Jacques Lacan (Encyclopædia Britannica Online)

Jacques LacanFrench psychologist in full Jacques Marie Émile Lacan

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French psychoanalyst who gained an international reputation as an original interpreter of Sigmund Freud’s work.

Lacan earned a medical degree in 1932 and was a practicing psychiatrist and psychoanalyst in Paris for much of his career. He helped introduce Freudian theory into France in the 1930s, but he reached prominence only after he began conducting regular seminars at the University of Paris in 1953. He acquired celebrity status in France after the publication of his essays and lectures in Écrits (1966; Eng. trans. The Language of the Self: The Function of Language in Psychoanalysis). He founded and headed an organization called the Freudian School of Paris from 1964 until he disbanded it in 1980 for what he claimed was its failure to adhere with sufficient strictness to Freudian principles.

Lacan emphasized the primacy of language as the mirror of the unconscious mind, and he tried to introduce the study of language (as practiced in modern linguistics, philosophy, and poetics) into psychoanalytic theory. His major achievement was his reinterpretation of Freud’s work in terms of the structural linguistics developed by French writers in the second half of the 20th century. The influence he gained extended well beyond the field of psychoanalysis to make him one of the dominant figures in French cultural life during the 1970s. In his own psychoanalytic practice, Lacan was known for his unorthodox, and even eccentric, therapeutic methods.

External Web sites

This topic is discussed at the following external Web sites.

European Graduate School – Biography of Jacques Lacan

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Jacques Lacan.” Encyclopædia Britannica. 2008. Encyclopædia Britannica Online. 19 Apr. 2008 <http://www.britannica.com/EBchecked/topic/327112/Jacques-Lacan>.

 

APA Style:

Jacques Lacan. (2008). In Encyclopædia Britannica. Retrieved April 19, 2008, from Encyclopædia Britannica Online: http://www.britannica.com/EBchecked/topic/327112/Jacques-Lacan

 

More from Britannica on “Jacques Lacan”
Jacques Lacan (French psychologist)

French psychoanalyst who gained an international reputation as an original interpreter of Sigmund Freud’s work.

Lacan earned a medical degree in 1932 and was a practicing psychiatrist and psychoanalyst in Paris for much of his career. He helped introduce Freudian theory into France in the 1930s, but he reached prominence only after he began conducting regular seminars at the University of Paris in 1953. He acquired celebrity status in France after the publication of his essays and lectures in Écrits (1966; Eng. trans. The Language of the Self: The Function of Language in Psychoanalysis). He founded and headed an organization called the Freudian School of Paris from 1964 until he disbanded it in 1980 for what he claimed was its failure to adhere with sufficient strictness to Freudian principles.

Lacan emphasized the primacy of language as the mirror of the unconscious mind, and he tried to introduce the study of language (as practiced in modern linguistics, philosophy, and poetics) into psychoanalytic theory. His major achievement was his reinterpretation of Freud’s work in terms of the structural linguistics developed by French writers in the second half of the 20th century. The influence he gained extended well beyond the field of psychoanalysis to make him one of the dominant figures in French cultural life during the 1970s. In his own psychoanalytic practice, Lacan was known for his unorthodox, and even eccentric, therapeutic methods.

Freudian School of Paris (French organization)

Aspects of this topic are discussed in the following places at Britannica.

  • Irigaray Irigaray, LuceIrigaray was a member of the Freudian School of Paris, founded by Jacques Lacan, and taught at the University of Paris VIII-Vincennes from 1968 until she was dismissed in 1974 because of her doctoral thesis. Entitled Speculum de l’autre femme (Speculum of the Other Woman), it argues that history and culture are written in patriarchal language, that they exclude women’s needs and…
  • Lacan Lacan, Jacques…of his essays and lectures in Écrits (1966; Eng. trans. The Language of the Self: The Function of Language in Psychoanalysis). He founded and headed an organization called the Freudian School of Paris from 1964 until he disbanded it in 1980 for what he claimed was its failure to adhere with sufficient strictness to Freudian principles.
Tel Quel (French journal)

French avant-garde literary review published from 1960 to 1982 by Éditions du Seuil. Founded by Philippe Sollers and other young writers, this eclectic magazine published works by such practitioners of the nouveau roman (“new novel”) as Alain Robbe-Grillet and Nathalie Sarraute, as well as works by these writers’ acknowledged predecessors— e.g., James Joyce and Francis Ponge.

Much influenced by Surrealism, Tel Quel had as a goal the evaluation of 20th-century literature; it printed previously unpublished works by Antonin Artaud, Georges Bataille, and Ezra Pound, as well as contemporary literary criticism by Michel Foucault, Jacques Derrida, Julia Kristeva, Roland Barthes, and Jacques Lacan. From 1966 to 1970 Tel Quel represented a Maoist view of Marxism.

From 1974 the review relinquished political involvement, becoming a supporter of such intellectuals as Bernard-Henri Lévy and André Glucksmann and others in the “new philosophers” movement. The critical orientation of Tel Quel shifted toward the classical Greco-Hebrew tradition, including discussion of biblical and theological questions. Its new stance included unequivocal support of worldwide human rights and the beginnings of an appreciation of modern culture, particularly that of the United States.

Aspects of this topic are discussed in the following places at Britannica.

  • place in French literature French literature…style of filmmaking. The nouveau roman was taken up by the literary theorist Jean Ricardou and promulgated by him through the avant-garde critical journal Tel Quel. (Founded in 1960 by Philippe Sollers and other writers, Tel Quel reflects the transformation and politicization of Parisian and…
  • significance to Sarduy Sarduy, Severo…fearful of its persecution of homosexuals and the censorship imposed on writers, Sarduy never went home. In Paris he became close to the group of critics and theoreticians…
Jean-François Revel (French philosopher and journalist)

French philosopher and journalist (b. Jan. 19, 1924, Marseille, France—d. April 30, 2006, Kremlin-Bicêtre, near Paris, France), was a defender of American liberal democracy, an unusual position for a French intellectual. Ricard adopted the pen name Revel in the Resistance during World War II. He graduated in philosophy from the École Normale Supérieure in 1943, after which he taught in Algeria, Mexico, and Italy before returning to France in 1956. His polemical Pourquoi des philosophes? (1957) criticized Marxism, the German philosopher Martin Heidegger, and the French psychoanalyst Jacques Lacan. In 1963 Revel left teaching for full-time journalism. Among his most influential books were Ni Marx, ni Jésus (1970; Without Marx or Jesus, 1971), La Tentation totalitaire (1976, The Totalitarian Temptation, 1977), Comment les démocraties finissent (1983; How Democracies Perish, 1984), and L’Obsession anti-américaine (2002; Anti-Americanism, 2003). In 1997 Revel was elected to the Académie Française.

semiotics (study of signs)

the study of signs and sign-using behaviour. It was defined by one of its founders, the Swiss linguist Ferdinand de Saussure, as the study of “the life of signs within society.” Although the word was used in this sense in the 17th century by the English philosopher John Locke, the idea of semiotics as an interdisciplinary mode for examining phenomena in different fields emerged only in the late 19th and early 20th centuries with the independent work of Saussure and of the American philosopher Charles Sanders Peirce.

Peirce’s seminal work in the field was anchored in pragmatism and logic. He defined a sign as “something which stands to somebody for something,” and one of his major contributions to semiotics was the categorization of signs into three main types: (1) an icon, which resembles its referent (such as a road sign for falling rocks); (2) an index, which is associated with its referent (as smoke is a sign of fire); and (3) a symbol, which is related to its referent only by convention (as with words or traffic signals). Peirce also demonstrated that a sign can never have a definite meaning, for the meaning must be continuously qualified.

Saussure treated language as a sign-system, and his work in linguistics has supplied the concepts and methods that semioticians apply to sign-systems other than language. One such basic semiotic concept is Saussure’s distinction between the two inseparable components of a sign: the signifier, which in language is a set of speech sounds or marks on a page, and the signified, which is the concept or idea behind the sign. Saussure also distinguished parole, or actual individual utterances, from langue, the underlying system of conventions that makes such utterances understandable; it is this underlying langue that most interests semioticians.

This interest in the structure behind the use of particular…

 

Jacques Lacan

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Western Philosophy
20th-century philosophy
Name
Jacques Lacan
Birth April 13, 1901
Death September 9, 1981
School/tradition Psychoanalysis, Structuralism
Main interests Psychoanalysis
Notable ideas The Mirror Stage,
The Real,
The Symbolic,
The Imaginary
Influenced by Hegel, Saussure, Heidegger, Freud, Lévi-Strauss, Kojève
Influenced Guattari, Miller, Milner, Althusser, Žižek, Laclau, Mouffe, Irigaray, Badiou, Castoriadis, Lefort, Butler, McLuhan
Part of a series of articles on
Psychoanalysis
Psychoanalysis
Constructs
Psychosexual development
Psychosocial development
ConsciousPreconsciousUnconscious
Id, ego, and super-ego
LibidoDrive
TransferenceSublimationResistanceImportant Figures
Sigmund FreudCarl Jung
Alfred AdlerOtto Rank
Anna FreudMargaret Mahler
Karen HorneyJacques Lacan
Ronald FairbairnMelanie Klein
Harry Stack Sullivan
Erik EriksonNancy Chodorow
Susan Sutherland Isaacs
Ernest JonesHeinz Kohut
John Bowlby

Important works
The Interpretation of Dreams
Four Fundamental Concepts of Psychoanalysis
Beyond the Pleasure Principle
Civilization and Its Discontents

Schools of Thought
Self psychologyLacanian
Analytical psychologyObject relations
InterpersonalRelational
AttachmentEgo psychology

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Jacques-Marie-Émile Lacan (French pronounced [ʒak lakɑ̃]) (April 13, 1901September 9, 1981) was a French psychoanalyst, psychiatrist, and doctor, who made prominent contributions to the psychoanalytic movement. His yearly seminars, conducted in Paris from 1953 until his death in 1981, were a major influence in the French intellectual milieu of the 1960s and ’70s, particularly among post-structuralist thinkers.

Lacan’s ideas centered on Freudian concepts such as the unconscious, the castration complex, the ego, focusing on identifications, and the centrality of language to subjectivity. His work was interdisciplinary, drawing on linguistics, philosophy, mathematics, amongst others. Although a controversial and divisive figure, Lacan is widely read in critical theory, literary studies, and twentieth-century French philosophy, as well as in the living practice of clinical psychoanalysis.

[edit] Biography

Because Lacan, like Freud, destroyed most of his records, it is difficult to disentangle the myths, anecdotes, and rumors that have surrounded him.

[edit] Early life

Jacques Lacan was born in Paris, the eldest child of three born to Emilie and Alfred Lacan. Alfred was a successful, middle-class salesman dealing in soap and oils. Emilie was an ardent Catholic, and Lacan’s younger brother eventually entered monastic life in 1929. Lacan attended the Collège Stanislas, a well-known Jesuit high school. In the early 1920s, Lacan attended some meetings of right-wing group Action Française and met its founder Charles Maurras. By the mid-1920s, Lacan’s growing anti-religious sentiment led to tensions with his Catholic family.[1][2]

Too thin to be accepted into military service, Lacan went directly into medical school in 1920, specializing in psychiatry from 1926. He took his clinical training at Sainte-Anne, the major psychiatric hospital in central Paris. In his studies he had a particular interest in the philosophic work of Karl Jaspers and Martin Heidegger and, alongside many other Parisian intellectuals of the time, he attended the famous seminars on Hegel given by Alexandre Kojève.

Beginning in the 1920s, Lacan undertook analysis with Rudolph Loewenstein, which continued until 1938.

[edit] 1930s

In 1931 Lacan received his license as a forensic psychiatrist, and in 1932 was awarded the Doctorat d’état for his thesis, De la Psychose paranoiaque dans les rapports avec la personnalité. While this thesis drew considerable acclaim outside psychoanalytic circles, particularly among the surrealist artists, it was largely ignored by psychoanalysts. In 1934 Lacan became a candidate for the Société Psychoanalytique de Paris. In January of that year, he married Marie-Louise Blondin, who gave birth to their first child, Caroline, the same month. Another child, Thibaut, was born in August of 1939.

He presented his first analytic paper on the “Mirror Phase” at the 1936 Congress of the International Psychoanalytical Association in Marienbad. According to Roudinesco, Lacan’s reading was interrupted by chairman of the congress, Ernest Jones, who was unwilling to offer more than the alloted time. Frustrated with what he considered an insult, Lacan left the congress to witness first hand a mass event manipulated by Nazis, in the form of the Olympic Games in Berlin. No copy of Lacan’s original lecture remains extant.[3]

Lacan was very active in the world of Parisian writers, artists and intellectuals during the inter-war period. In addition to André Breton and Georges Bataille, he was also associated with Salvador Dalí and Pablo Picasso. He attended the mouvement Psyché founded by Maryse Choisy. Several of his early articles were published in the Surrealist journal Minotaure and he was present at the first public reading of James Joyce’s Ulysses. Dylan Evans has speculated that Lacan was a surrealist at heart, “his interest in surrealism predates his interest in psychoanalysis. Perhaps Lacan never really abandoned his early surrealist sympathies, its neo-Romantic view of madness as ‘convulsive beauty’, its celebration of irrationality, and its hostility to the scientist who murders nature by dissecting it.”[4] As such company would suggest, during this period Lacan was better known in literary circles than psychoanalytic ones.

[edit] 1940s

The Société Psychoanalytique de Paris (SPP) was disbanded due to Nazi Germany’s occupation of France in 1940 and Lacan was subsequently called up to serve in the French army at the Val-de-Grâce military hospital in Paris, where he spent the duration of the war. His third child, Sibylle, was born in 1940.

The following year, Lacan fathered a child, Judith (who kept the name Bataille) with Sylvia Bataille (née Maklès), estranged wife of his friend Georges. There are contradictory stories about his romantic life with Sylvia Bataille in southern France during World War II. The official record shows only that Marie-Louise requested divorce after Judith’s birth, and Lacan married Sylvia in 1953.

Following the war, the SPP recommenced their meetings, and Lacan visited England for a five-week study trip, meeting English analysts Wilfred Bion and John Rickman. He was influenced by Bion’s analytic work with groups and this contributed to his own later emphasis on study groups as a structure with which to advance theoretical work in psychoanalysis.

In 1949, Lacan presented a new paper on the mirror stage to the sixteenth IPA congress in Zurich.

[edit] 1950s

In 1951 Lacan started to hold a private weekly seminar in Paris, urging what he described as “a return to Freud” concentrating upon the linguistic nature of psychological symptomatology. Becoming public in 1953, Lacan’s twenty-seven year long seminar was very influential in Parisian cultural life as well as in psychoanalytic theory and clinical practice.

In 1953, after a disagreement about analytic practice methods, Lacan and many of his colleagues left the Société Parisienne de Psychoanalyse to form a new group the Société Française de Psychanalyse (SFP). One of the consequences of this was to deprive the new group of membership within the International Psychoanalytical Association.

Encouraged by the reception of “the return to Freud” and of his report – “The Function and Field of Speech and Language in Psychoanalysis” (Écrits) – Lacan again returned to Freud, re-reading the canon in relation with contemporary philosophy, linguistics, ethnology, biology and topology. From 1953 to 1964 at Sainte-Anne Hospital , he held his Seminars and presented case histories of patients. During this period he wrote the texts that are found in Écrits, a selection of which was first published in 1966. In his seventh Seminar of 1959-60, ‘The Ethics of Psychoanalysis’, Lacan defined his ethical foundations of psychoanalysis and constructs his “ethics for our time”; according to Freud, an ethics that would prove to be equal to the tragedy of modern man and to the “discontent of civilization”. At the roots of the ethics is desire: analysis’ only promise is austere, it is the entrance-into-the-I (in French a play of words between ‘l’entrée en je’ and ‘l’entrée en jeu’). ‘I must come to the place where the id was’, where the analysand discovers, in its absolute nakedness, the truth of his desire. The end of psychoanalysis entails ‘the purification of desire’. This text functions throughout the years as the background of Lacan’s work. He defends three assertions: that psychoanalysis must have a scientific status; that Freudian ideas have radically changed the concepts of subject, of knowledge, and of desire; that the analytic field is the only place from where it is possible to question the insufficiencies of science and philosophy.

[edit] 1960s

Starting in 1962 a complex negotiation took place to determine the status of the SFP within the IPA. Lacan’s practice—with his controversial indeterminate-length sessions in which he charged a full fee for truncated sessions, had his hair cut during sessions,[5] and Lacan’s critical stance towards psychoanalytic orthodoxy—led, in 1963, to a condition being set by the IPA that registration of the SFP was dependent upon removing Lacan from the list of SFP training analysts. Lacan left the SFP to form his own school which became known as the École Freudienne de Paris (EFP)

With Lévi-Strauss and Althusser‘s support, he was appointed lecturer at the École Pratique des Hautes Etudes. He started with a seminar on The Four Fundamental Concepts of Psychoanalysis in January 1964 in the Dussane room at the École Normale Supérieure (in his first session he thanks the generosity of Fernand Braudel and Claude Lévi-Strauss). Lacan began to set forth his own teaching on psychoanalysis to an audience of colleagues who had joined him from the SFP. His lectures also attracted many of the École Normale’s students. He divided the École de la Cause freudienne into three sections: the section of pure psychoanalysis (training and elaboration of the theory, where members who have been analyzed but haven’t become analysts can participate); the section for applied psychoanalysis (therapeutic and clinical, physicians who have neither completed nor started analysis are welcome); the section for taking inventory of the Freudian field (it concerned the critique of psychoanalytic literature and the analysis of the theoretical relations with related or affiliated sciences (Proposition du 9 octobre 1967 sur le psychanalyste à l’Ecole).

By the 1960s, Lacan was associated—at least in the public mind—with the far left in France.[6] In May 1968 Lacan voiced his sympathy for the student protests and as a corollary a Department of Psychology was set up by his followers at the University of Vincennes (Paris VIII). Echoing this sentiment, “Shortly after the tumultuous events of May 1968, Lacan was accused by the authorities of being a subversive, and directly influencing the events that transpired.”[7]

In 1969 Lacan moved his public seminars to the Faculté de Droit (Panthéon) where he continued to deliver his expositions of analytic theory and practice till the dissolution of his School in 1980.

[edit] 1970s

Throughout the final decade of his life, Lacan continued his widely followed seminars. During this period, he focuses on the development of his concepts of masculine and feminine jouissance, and puts special emphasis on his concept of “The Real” as a point of impossible contradiction in the “Symbolic Order”. This late work had the greatest influence on feminist thought, as well as upon the informal movement that arose in the 1970s or 1980s called post-modernism.

[edit] Major concepts

[edit] The ‘Return to Freud’

Lacan’s “return to Freud” emphasizes a renewed attention to the original texts of Freud and a radical critique of Ego psychology, Melanie Klein and Object relations theory. Lacan thought that Freud’s ideas of “slips of the tongue”, jokes, etcetera, all emphasized the agency of language in subjective constitution. “Correcting” Freud from within the light of Saussure, Lévi-Strauss, and Barthes, Lacan’s “return to Freud” could be read as the realization that the pervading agency of the unconscious is intimately tied to the functions and dynamics of language, where the signifier is irremediably divorced from the signified in a chronic but generative tension of lack. In “The Instance of the Letter in the Unconscious, or Reason Since Freud” (Écrits, pp. 161 – 197).) he argued that “the unconscious is structured like a language”; it was not a primitive or archetypal part of the mind separate from the conscious, linguistic ego, but a formation as complex and structurally sophisticated as consciousness itself. If the unconscious is structured like a language, he claimed, then the self is denied any point of reference to which to be ‘restored’ following trauma or ‘identity crisis’.

[edit] The mirror stage (le stade du miroir)

Lacan’s first official contribution to psychoanalysis was the mirror stage which he described “… as formative of the function of the I as revealed in psychoanalytic experience”. By the early fifties, he no longer considers the mirror stage as only a moment in the life of the infant, but as the permanent structure of subjectivity. In the paradigm of The Imaginary order, the subject is permanently caught and captivated by his own image. Lacan writes, “[T]he mirror stage is a phenomenon to which I assign a twofold value. In the first place, it has historical value as it marks a decisive turning-point in the mental development of the child. In the second place, it typifies an essential libidinal relationship with the body-image”.[8]

As he further develops the concept, the stress falls less on its historical value and ever more on its structural value.[4] In his fourth Seminar, La relation d’objet, Lacan states that “the mirror stage is far from a mere phenomenon which occurs in the development of the child. It illustrates the conflictual nature of the dual relationship”.

The mirror stage describes the formation of the Ego via the process of objectification, the Ego being the result of feeling dissention between one’s perceived visual appearance and one’s perceived emotional reality. This identification is what Lacan called alienation. At six months the baby still lacks coordination, however, he can recognize himself in the mirror before attaining control over his bodily movements. He sees his image as a whole, and the synthesis of this image produces a sense of contrast with the uncoordination of the body, which is perceived as a fragmented body. This contrast is first felt by the infant as a rivalry with his own image, because the wholeness of the image threatens him with fragmentation, and thus the mirror stage gives rise to an aggressive tension between the subject and the image. To resolve this aggressive tension, the subject identifies with the image: this primary identification with the counterpart is what forms the Ego.[4] The moment of identification is to Lacan a moment of jubilation since it leads to an imaginary sense of mastery, yet the jubilation may also be accompanied by a depressive reaction, when the infant compares his own precarious sense of mastery with the omnipotence of the mother.[9] This identification also involves the ideal ego which functions as a promise of future wholeness sustaining the Ego in anticipation.

In the Mirror stage a misunderstanding – “méconnaissance” – constitutes the Ego–the ‘moi’ becomes alienated from himself through the introduction of the Imaginary order subject. It must be said that the mirror stage has also a significant symbolic dimension. The Symbolic order is present in the figure of the adult who is carrying the infant: the moment after the subject has jubilantly assumed his image as his own, he turns his head towards this adult who represents the big Other, as if to call on him to ratify this image.[10]

[edit] Other/other

While Freud uses the term “other”, referring to der Andere (the other person) and “das Andere” (otherness), Lacan’s use is more like Hegel’s, through Alexandre Kojève.

Lacan often used an algebraic symbology for his concepts:[11] the big Other is designated A (for French Autre) and the little other is designated a (italicized French autre). He asserts that an awareness of this distinction is fundamental to analytic practice: ‘the analyst must be imbued with the difference between A and a,[12] so he can situate himself in the place of Other, and not the other’.[13]

  1. The little other is the other who is not really other, but a reflection and projection of the Ego. He is both the counterpart or the other people in whom the subject perceives a visual likeness (semblable), and the specular image or the reflection of one’s body in the mirror. In this way the little other is entirely inscribed in The Imaginary order. See Objet Petit a.
  2. The big Other designates a radical alterity, an otherness transcending the illusory otherness of the Imaginary because it cannot be assimilated through identification. Lacan equates this radical alterity with language and the law: the big Other is inscribed in The Symbolic order, being in fact the Symbolic insofar as it is particularized for each subject. The Other is then another subject and also the Symbolic order which mediates the relationship with that other subject.

‘The Other must first of all be considered a locus, the locus in which speech is constituted’.[14] We can speak of the Other as a subject in a secondary sense, only when a subject may occupy this position and thereby embody the Other for another subject.[15]

When he argues that speech originates not in the Ego nor in the subject, but in the Other, Lacan stresses that speech and language are beyond one’s conscious control; they come from another place, outside consciousness, and then ‘the unconscious is the discourse of the Other’.[16] When conceiving the Other as a place, Lacan refers to Freud’s concept of physical locality, in which the unconscious is described as “the other scene”.

“It is the mother who first occupies the position of the big Other for the child, it is she who receives the child’s primitive cries and retroactively sanctions them as a particular message”.[4] The castration complex is formed when the child discovers that this Other is not complete, that there is a Lack (manque) in the Other. This means that there is always a signifier missing from the trove of signifiers constituted by the Other. Lacan illustrates this incomplete Other graphically by striking a bar through the symbol A; hence another name for the castrated, incomplete Other is the ‘barred Other’.[17][18]

[edit] The Three Orders

[edit] The Imaginary

Lacan thought the relationship between the Ego and the reflected image means that the Ego and the Imaginary order itself are places of radical alienation: “alienation is constitutive of the Imaginary order”.[14] This relationship is also narcissistic. So the Imaginary is the field of images and imagination, and deception: the main illusions of this order are synthesis, autonomy, duality, similarity.

The Imaginary is structured by the Symbolic order: in The Four Fundamental Concepts of Psychoanalysis Lacan argues how the visual field is structured by symbolic laws. Thus the Imaginary involves a linguistic dimension. If the signifier is the foundation of the Symbolic, the signified and signification are part of the Imaginary order. Language has Symbolic and Imaginary connotations; in its Imaginary aspect, language is the “wall of language” which inverts and distorts the discourse of the Other. On the other hand, the Imaginary is rooted in the subject’s relationship with its own body (the image of the body). In Fetishism: the Symbolic, the Imaginary and the Real Lacan argues that in the sexual plane the Imaginary appears as sexual display and courtship love.

Lacan accused major psychoanalytic schools of reducing the practice of psychoanalysis to the Imaginary order by making identification with the analyst the objective of analysis (see Écrits, “The Directions of the Treatment”). He proposes the use of the Symbolic as the way to dislodge the disabling fixations of the Imaginary: the analyst transforms the images into words. “The use of the Symbolic is the only way for the analytic process to cross the plane of identification.”[19]

[edit] The Symbolic

In his Seminar IV “La relation d’objet” Lacan asserts that the concepts of Law and Structure are unthinkable without language: thus the Symbolic is a linguistic dimension. Yet, he does not simply equate this order with language since language involves the Imaginary and the Real as well. The dimension proper of language in the Symbolic is that of the signifier, that is a dimension in which elements have no positive existence but which are constituted by virtue of their mutual differences.

The Symbolic is also the field of radical alterity, that is the Other: the unconscious is the discourse of this Other. Besides it is the realm of the Law which regulates desire in the Oedipus complex. We may add that the Symbolic is the domain of culture as opposed to the Imaginary order of nature. As important elements in the Symbolic, the concepts of death and lack (manque) connive to make of the pleasure principle the regulator of the distance from the Thing (das ding an sich) and the death drive which goes “beyond the pleasure principle by means of repetition”—”the death drive is only a mask of the Symbolic order.”[11]

It is by working in the Symbolic order that the analyst can produce changes in the subjective position of the analysand; these changes will produce imaginary effects since the Imaginary is structured by the Symbolic.[4] Thus, it is the Symbolic which is determinant of subjectivity, and the Imaginary, made of images and appearances, is the effect of the Symbolic.

[edit] The Real

Not only opposed to the Imaginary, the Real is also located outside the Symbolic. Unlike the latter which is constituted in terms of oppositions, i.e. presence/absence, “there is no absence in the Real.”[11] Whereas the Symbolic opposition presence/absence implies the possibility that something may be missing from the Symbolic, “the Real is always in its place.”[19] If the Symbolic is a set of differentiated elements, signifiers, the Real in itself is undifferentiated, it bears no fissure. The Symbolic introduces “a cut in the real”, in the process of signification: “it is the world of words that creates the world of things – things originally confused in the “here and now” of the all in the process of coming into being.[20]

Thus the Real is that which is outside language, resisting symbolization absolutely. In Seminar XI Lacan defines the Real as “the impossible” because it is impossible to imagine and impossible to integrate into the Symbolic, being impossibly attainable. It is this resistance to symbolization that lends the Real its traumatic quality. In his Seminar “La relation d’objet”, Lacan reads Freud’s case on “Little Hans.” He distinguishes two real elements which intrude and disrupt the child’s imaginary pre-oedipical harmony: the real penis which is felt in infantile masturbation and the newly born sister.

Finally, the Real is the object of anxiety in that it lacks any possible mediation, and is “the essential object which is not an object any longer, but this something faced with which all words cease and all categories fail, the object of anxiety par excellence.”[11]

[edit] Desire

Lacan’s désir follows Freud’s concept of Wunsch and it is central to Lacanian theories. For the aim of the talking cure – psychoanalysis – is precisely to lead the analysand to uncover the truth about their desire, but this is only possible if that desire is articulated, or spoken.[21] Lacan said that “it is only once it is formulated, named in the presence of the other, that desire appears in the full sense of the term.”[22] “That the subject should come to recognize and to name his/her desire, that is the efficacious action of analysis. But it is not a question of recognizing something which would be entirely given. In naming it, the subject creates, brings forth, a new presence in the world.”[11] “[W]hat is important is to teach the subject to name, to articulate, to bring desire into existence.” Now, although the truth about desire is somehow present in discourse, discourse can never articulate the whole truth about desire: whenever discourse attempts to articulate desire, there is always a leftover, a surplus.[21]

In The Signification of the Phallus Lacan distinguishes desire from need and demand. Need is a biological instinct that is articulated in demand, yet demand has a double function, on one hand it articulates need and on the other acts as a demand for love. So, even after the need articulated in demand is satisfied, the demand for love remains unsatisfied and this leftover is desire.[23] For Lacan “desire is neither the appetite for satisfaction nor the demand for love, but the difference that results from the subtraction of the first from the second” (article cited). Desire then is the surplus produced by the articulation of need in demand (Dylan Evans). Lacan adds that “desire begins to take shape in the margin in which demand becomes separated from need” (article cited). Hence desire can never be satisfied, or as Slavoj Žižek puts it “desire’s raison d’être is not to realize its goal, to find full satisfaction, but to reproduce itself as desire.”

It is also important to distinguish between desire and the drives. If they belong to the field of the Other (as opposed to love), desire is one, whereas the drives are many. The drives are the partial manifestations of a single force called desire (see “The Four Fundamental Concepts of Psychoanalysis”). If one can surmise that objet petit a is the object of desire, it is not the object towards which desire tends, but the cause of desire. For desire is not a relation to an object but a relation to a lack (manque). Then desire appears as a social construct since it is always constituted in a dialectical relationship.

[edit] Drives

Lacan maintains Freud’s distinction between Trieb (drive) and Instinkt (instinct) in that drives differ from biological needs because they can never be satisfied and do not aim at an object but rather circle perpetually round it, so the real source of jouissance is to repeat the movement of this closed circuit. In the same Seminar Lacan posits the drives as both cultural and symbolic (discourse) constructs, to him “the drive is not a given, something archaic, primordial”. Yet he incorporates the four elements of the drives as defined by Freud (the pressure, the end, the object and the source) to his theory of the drive’s circuit: the drive originates in the erogenous zone, circles round the object, and then returns to the erogenous zone. The circuit is structured by the three grammatical voices:

  1. the active voice (to see)
  2. the reflexive voice (to see oneself)
  3. the passive voice (to be seen)

The active and reflexive voices are autoerotic, they lack a subject. It is only when the drive completes its circuit with the passive voice that a new subject appears. So although it is the “passive” voice, the drive is essentially active, “to make oneself be seen” instead of “to be seen.” The circuit of the drive is the only way for the subject to transgress the pleasure principle.

Lacan identifies four partial drives: the oral drive (the erogenous zones are the lips, the partial object the breast), the anal drive (the anus and the faeces), the scopic drive (the eyes and the gaze) and the invocatory drive (the ears and the voice). The first two relate to demand and the last two to desire. If the drives are closely related to desire, they are the partial aspects in which desire is realized: again, desire in one and undivided whereas the drives are partial manifestations of desire.

[edit] Other concepts

[edit] Writings and seminars

Although Lacan is a major figure in the history of psychoanalysis, his Seminar lectures – contains the majority of his life’s work, though some of these remain yet unpublished. Jacques-Alain Miller, the sole editor of Lacan’s seminars, has been regularly conducting since 1984 a series of lectures, “L’orientation lacanienne”, within the structure of ParisVIII. Miller’s teachings have been published in the US by the journal Lacanian Ink.

Lacan claims that his Écrits were not to be understood, but would produce a meaning effect in the reader similar to some mystical texts. Lacan’s writing is notoriously difficult due to the repeated Hegelian/Kojèvean allusions, wide theoretical divergences from other psychoanalytic and philosophical thinking, and Lacan’s obscure prose style.

[edit] Criticism

Although Lacan is associated with it, he was criticized by major figures associated with post-structuralism (and the related postmodernism school). Jacques Derrida characterized Lacan as taking a structuralist approach to psychoanalysis. Derrida claimed this led Lacan to inherit a Freudian “phallocentrism,” exemplified by Lacan’s conception of the phallus as the “primary signifier” that determines the social order of signifiers. Derrida deconstructs the Freudian conception of “penis envy”, upon which female subjectivity is determined “as an absence,” to show that the primacy of the male phallus entails a hierarchy between phallic presence and absence that ultimately collapses.

While he has been criticized for adopting a Freudian phallocentric stance in his psychoanalytic theories, many feminists believe Lacan provides a useful analysis of gender biases and imposed roles. Some feminist critics, such as Luce Irigaray,[24] accuse Lacan of maintaining the sexist tradition in psychoanalysis. Others feminists, such as Judith Butler,[25] Jane Gallop,[26] Bracha Ettinger, [27] and Elizabeth Grosz,[7] have each interpreted Lacan’s work as opening up new possibilities for feminist theory.

Other critics have often dismissed Lacan and his work in a more-or-less wholesale fashion. François Roustang[28] called Lacan’s output “extravagant” and an “incoherent system of pseudo-scientific gibberish.” Noam Chomsky described Lacan as “an amusing and perfectly self-conscious charlatan”[29].In Fashionable Nonsense, Alan Sokal and Jean Bricmont accuse Lacan of “superficial erudition” and of abusing scientific concepts he does not understand.

Defenders of Lacanian thinking dispute most external criticism, stating that these critics’ misunderstand—or often simply have not read—Lacan’s texts. Bruce Fink has dismissed Sokal and Bricmont, claiming they have “no idea whatsoever what Lacan is up to,” and accuses them of elevating a distaste for Lacan’s writing style into an attack on his thought as a whole.[30] Similarly, Arkady Plotnitsky claims that Lacan uses the mathematical concepts more accurately than do Sokal and Bricmont.[31]

[edit] Sources

[edit] References

  1. ^ Roudinesco, Elisabeth Jacques Lacan & Co.: a history of psychoanalysis in France, 1925-1985, 1990, Chicago University Press
  2. ^ Perry Meisel (April 13, 1997). The Unanalyzable. New York Times.
  3. ^ Roudinesco, Elisabeth. ‘The mirror stage: an obliterated archive’ The Cambridge Companion to Lacan. Ed. Jean-Michel Rabaté. Cambridge: CUP, 2003
  4. ^ a b c d e Evans, Dylan “From Lacan to Darwin”, in The Literary Animal; Evolution and the Nature of Narrative, eds. Jonathan Gottschall and David Sloan Wilson, Evanston: Northwestern University Press, 2005
  5. ^ Review of Bruce Fink’s A Clinical Introduction to Lacanian Psychoanalysis: Theory and Technique
  6. ^ French Communist Party “official philosopher” Louis Althusser did much to advance this association in the 1960s. Zoltán Tar and Judith Marcus in Frankfurt school of sociology. ISBN 0878559639 (p.276) write, for example, Althusser’s call to Marxists that the Lacanian enterprise might … help further revolutionary ends, endorsed Lacan’s work even further.
  7. ^ a b Elizabeth A. Grosz, Jacques Lacan: A Feminist Introduction
  8. ^ Lacan, J., Some reflections on the Ego in Écrits
  9. ^ Lacan, J., La relation d’objet in Écrits
  10. ^ Lacan, Tenth Seminar, L’angoisse, 1962-1963
  11. ^ a b c d e Lacan, J., The Seminar of Jacques Lacan: Book II : The Ego in Freud’s Theory and in the Technique of Psychoanalysis 1954-1955 (W. W. Norton & Company, 1991), ISBN 9780393307092
  12. ^ Lacan, J., The Freudian Thing in Écrits
  13. ^ Lacan, J., Psychoanalysis and its Teaching in Écrits
  14. ^ a b Lacan, Seminar III: The Psychoses
  15. ^ Lacan, Seminar VIII: Le transfert
  16. ^ Lacan, J., Seminar on “The Purloined Letter” in Écrits
  17. ^ Lacan, J., The Agency of the Letter in the Unconscious in Écrits
  18. ^ Lacan, Seminar V: Les formations de l’inconscient
  19. ^ a b Lacan, J. Seminar XI: The Four Fundamental Concepts of Psychoanalysis
  20. ^ Lacan, J., The Function and Field of Speech and Language in Psychoanalysis in Écrits
  21. ^ a b Fink, Bruce, The Lacanian Subject: Between Language and Jouissance (Princeton University Press, 1996), ISBN 9780691015897
  22. ^ Lacan, J., The Seminar of Jacques Lacan: Book I : Freud’s Papers on Technique 1953-1954 “…what is important is to teach the subject to name, to articulate, to bring desire into existence” (W. W. Norton & Company, 1991), ISBN 9780393306972
  23. ^ Lacan, J., ‘The Signification of the Phallus’ in Écrits
  24. ^ Irigary, Luce, This Sex Which Is Not One 1977, (Eng. trans. 1985)
  25. ^ Butler, Judith. Bodies That Matter: On the Discursive Limits of “Sex” (1993)
  26. ^ Gallop, Jane, Reading Lacan. Ithaca: Cornell University Press, 1985.
  27. ^ Ettinger, Bracha L., The Matrixial Borderspace, University of Minnesota Press, 2006 (essays from 1994-1999, published in French as “Régard et éspace-de-bord matrixiels”, Bruxelles: La Lettre Volée, 1999) and Special Issue of Theory, Culture and Society, Vol. 21, n.1, 2004.
  28. ^ Roustang, François, The Lacanian Delusion
  29. ^ Usenet, 1996
  30. ^ Bruce Fink, Lacan to the Letter
  31. ^ Arkady Plotnitsky, The Knowable and the Unknowable

[edit] Bibliography

Selected works published in English listed below. More complete listings can be found at Lacan Dot Com.

*referenced above

[edit] Works about Lacan’s Work and Theory

  • Badiou, Alain, “The Formulas of L’Etourdit” (New York: Lacanian Ink 27, 2006.)
  • —————, “Lacan and the Pre-Socratics”, Lacan Dot Com, 2006.
  • Benvenuto, Bice; Kennedy, Roger, The Works of Jacques Lacan (London, 1986, Free Association Books.)
  • Bowie, Malcolm, Lacan (London: Fontana, 1991). (An introduction.)
  • Dor, Joel, The Clinical Lacan (New York: Other Press, 1999)
  • —————, Introduction to the Reading of Lacan: The Unconscious Structured Like a Language (New York: Other Press, 2001)
  • Elliott, Anthony and Stephen Frosh(eds.), Psychoanalysis in Contexts: Paths between Theory and Modern Culture (London and New York: Routledge, 1995). (A recent overview.)
  • Ettinger, Bracha L., “The Feminine/Prenatal Weaving in the Matrixial Subjectivity-as-Encounter.” Psychoanalytic Dialogues, VII:3, The Analytic Press, New York, 1997.
  • —————, “Matrixial Gaze and Screen: Other than Phallic and Beyond the Late Lacan.” In: Laura Doyle (ed.) Bodies of Resistance. Evanston, Illinois: Northwestern University Press, 2001.
  • —————, “Weaving Trans-Subjective Texture or The Matrixial Sinthome.” In : Thurston, Luke (ed.), Re-inventing the Symptom: Essays on the final Lacan. NY: The Other Press, 2002.
  • Evans, Dylan, An Introductory Dictionary of Lacanian Psychoanalysis, Routledge, 1996.
  • Fink, Bruce, The Lacanian Subject: Between Language and Jouissance (Princeton: Princeton University Press, 1995).
  • —————, Lacan to the Letter: Reading Ecrits Closely, University of Minnesota, 2004.
  • Forrester, John, Language and the Origins of Psychoanalysis (Basingstoke and London: Macmillan, 1985).
  • Fryer, David Ross, The Intervention of the Other: Ethical Subjectivity in Levinas and Lacan (New York: Other Press, 2004)
  • Gallop, Jane, Reading Lacan. Ithaca: Cornell University Press, 1985.
  • —————, The Daughter’s Seduction: Feminism and Psychoanalysis. Ithaca: Cornell University Press, 1982.
  • Gherovici, Patricia, The Puerto Rican Syndrome (New York: Other Press, 2003)
  • Glynos, Jason and Yannis Stravrakakis, ED, Lacan and Science. London :Karnac Books, May 2002.
  • Harari, Roberto, Lacan’s Four Fundamental Concepts of Psychoanalysis: An Introduction (New York: Other Press, 2004)
  • —————, Lacan’s Seminar on “Anxiety”: An Introduction (New York: Other Press, 2005)
  • Homer, Sean, Jacques Lacan (London: Routledge, 2005)
  • Lander, Romulo, Subjective Experience and the Logic of the Other (New York: Other Press, 2006)
  • Leupin, Alexandre, Lacan Today (New York: Other Press, 2004)
  • Mathelin, Catherine, Lacanian Psychotherpay with Children: The Broken Piano (New York: Other Press, 1999)
  • McGowan, Todd and Sheila Kunkle Eds., Lacan and Contemporary Film (New York: Other Press, 2004)
  • Miller, Jacques-Alain, “Introduction to Reading Jacques Lacan’s Seminar on Anxiety I ” (New York: Lacanian Ink 26, 2005.)
  • —————, “Introduction to Reading Jacques Lacan’s Seminar on Anxiety II” (New York: Lacanian Ink 27, 2006.)
  • —————, “Jacques Lacan’s Later Teachings” (New York: Lacanian Ink 21, 2003.)
  • —————, “The Paradigms of Jouissance” (New York, Lacanian Ink 17, 2000.)
  • —————, “Suture: Elements of the Logic of the Signifier”, Lacan Dot Com, 2006.
  • Moustafa, Safouan, Four Lessons of Psychoanalysis (New York: Other Press, 2004)
  • Nasio, Juan-David , Book of Love and Pain: The Thinking at the Limit with Freud and Lacan. Translated by David Pettigrew and Francois Raffoul (Albany: SUNY Press, 2003)
  • —————, Five Lessons on the Psychoanalytic Theory of Jacques Lacan. Translated by David Pettigrew and Francois Raffoul (Albany: SUNY Press, 1998)
  • —————, Hysteria: The Splendid Child of Psychoanalysis. Translated by Susan Fairfield (New York: Other Press, 1998)
  • Pettigrew, David and François Raffoul (eds.), Disseminating Lacan (Albany: SUNY Press, 1996)
  • Rabaté, Jean-Michel (ed.), The Cambridge Companion to Lacan (Cambridge: Cambridge University Press, 2003)
  • Rose, Jacqueline, Sexuality in the Field of Vision (London: Verso, 1986)
  • Roudinesco, Élisabeth, “Jacques Lacan: His Life and Work”. Translated by Bray B. New York, Columbia University Press, 1997
  • Turkle, Sherry, Psychoanalytic Politics: Jacques Lacan and Freud’s French Revolution, 2nd edition, Guildford Press, New York, 1992
  • ————— and Wollheim, Richard, ‘Lacan: an exchange’, New York Review of Books, 26 (9), 1979, p. 44.
  • Soler, Colette, What Lacan Said About Women Translated by John Holland (New York: Other Press, 2006)
  • Thurston, Luke (ed.), “Re-inventing the Symptom”, NY: Other Press, 2002.
  • Van Haute, Philippe, Against Adaptation: Lacan’s “Subversion” of the Subject (New York: Other Press, 2002)
  • Van Haute, Philippe and Tomas Geyskens, Confusion of Tongues: The Primacy of Sexuality in Freud, Ferenczi, and Laplanche (New York: Other Press, 2004)
  • Webster, Richard, Why Freud Was Wrong-Sin, Science and Psychoanalysis (Harper Collins, 1995.)
  • Wilden, Anthony, ‘Jacques Lacan: A partial bibliography’, Yale French Studies, 36/37, 1966, pp. 263–268.
  • Žižek, Slavoj, “Woman is One of the Names-of-the-Father, or how Not to misread Lacan´s formulas of sexuation”, Lacan Dot Com, 2005.
  • —————, ‘The object as a limit of discourse: approaches to the Lacanian real’, Prose Studies, 11 (3), 1988, pp. 94–120.
  • —————, Interrogating the Real, ed. Rex Butler and Scott Stephens (London and New York: Continuum, 2005).
  • —————, “Jacques Lacan as Reader of Hegel” (New York: Lacanian Ink 27, 2006.)

[edit] External links

[edit] Introductions

[edit] Practice

[edit] Theory

[edit] Criticism

 

Persondata
NAME Lacan, Jacques
ALTERNATIVE NAMES  
SHORT DESCRIPTION French psychologist
DATE OF BIRTH 13 April 1901
PLACE OF BIRTH Paris, France
DATE OF DEATH 9 September 1981
PLACE OF DEATH Paris, France

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From Logic to Ontology: The limit of “The Semantic Web”

 

 

(Some post are written in English and Spanish language) 

http://www.linkedin.com/answers/technology/web-development/TCH_WDD/165684-18926951 

From Logic to Ontology: The limit of “The Semantic Web” 

 http://en.wikipedia.org/wiki/Undecidable_problem#Other_problems

If you read the next posts on this blog: 

Semantic Web

The Semantic Web

What is the Semantic Web, Actually?

The Metaweb: Beyond Weblogs. From the Metaweb to the Semantic Web: A Roadmap

Semantics to the people! ontoworld

What’s next for the Internet

Web 3.0: Update

How the Wikipedia 3.0: The End of Google? article reached 2 million people in 4 days!

Google vs Web 3.0

Google dont like Web 3.0 [sic] Why am I not surprised?

Designing a better Web 3.0 search engine

From semantic Web (3.0) to the WebOS (4.0)

Search By Meaning

A Web That Thinks Like You

MINDING THE PLANET: THE MEANING AND FUTURE OF THE SEMANTIC WEB

The long-promised “semantic” web is starting to take shape

Start-Up Aims for Database to Automate Web Searching

Metaweb: a semantic wiki startup

http://www.freebase.com/

The Semantic Web, Collective Intelligence and Hyperdata.

Informal logic 

Logical argument

Consistency proof 

Consistency proof and completeness: Gödel’s incompleteness theorems

Computability theory (computer science): The halting problem

Gödel’s incompleteness theorems: Relationship with computability

Non-formal or Inconsistency Logic: LACAN’s LOGIC. Gödel’s incompleteness theorems,

You will realize the internal relationship between them linked from Logic to Ontology.  

I am writing from now on an article about the existence of the semantic web.  

I will prove that it does not exist at all, and that it is impossible to build from machines like computers.  

It does not depend on the software and hardware you use to build it: You cannot do that at all! 

You will notice the internal relations among them, and the connecting thread is the title of this post: “Logic to ontology.”   

I will prove that there is no such construction, which can not be done from the machines, and that does not depend on the hardware or software used.  

More precisely, the limits of the semantic web are not set by the use of machines themselves and biological systems could be used to reach this goal, but as the logic that is being used to construct it does not contemplate the concept of time, since it is purely formal logic and metonymic lacks the metaphor, and that is what Gödel’s theorems remark, the final tautology of each construction or metonymic language (mathematical), which leads to inconsistencies. 

This consistent logic is completely opposite to the logic that makes inconsistent use of time, inherent of human unconscious, but the use of time is built on the lack, not on positive things, it is based on denials and absences, and that is impossible to reflect on a machine because of the perceived lack of the required self-awareness is acquired with the absence.  

The problem is we are trying to build an intelligent system to replace our way of thinking, at least in the information search, but the special nature of human mind is the use of time which lets human beings reach a conclusion, therefore does not exist in the human mind the halting problem or stop of calculation.  

So all efforts faced toward semantic web are doomed to failure a priori if the aim is to extend our human way of thinking into machines, they lack the metaphorical speech, because only a mathematical construction, which will always be tautological and metonymic, and lacks the use of the time that is what leads to the conclusion or “stop”.  

As a demonstration of that, if you suppose it is possible to construct the semantic web, as a language with capabilities similar to human language, which has the use of time, should we face it as a theorem, we can prove it to be false with a counter example, and it is given in the particular case of the Turing machine and “the halting problem”.  

Then as the necessary and sufficient condition for the theorem is not fulfilled, we still have the necessary condition that if a language uses time, it lacks formal logic, the logic used is inconsistent and therefore has no stop problem.

This is a necessary condition for the semantic web, but it is not enough and therefore no machine, whether it is a Turing Machine, a computer or a device as random as a black body related to physics field, can deal with any language other than mathematics language hence it is implied that this language is forced to meet the halting problem, a result of Gödel theorem.   

De la lógica a la ontología: El límite de la “web semántica”  

Si lee los siguientes artículos de este blog: 

http://es.wikipedia.org/wiki/Web_sem%C3%A1ntica  

Wikipedia 3.0: El fin de Google (traducción Spanish)

Lógica 

Lógica Consistente y completitud: Teoremas de la incompletitud de Gödel (Spanish)

Consistencia lógica (Spanish)

Teoría de la computabilidad. Ciencia de la computación.

Teoremas de la incompletitud de Gödel y teoría de la computación: Problema de la parada 

Lógica inconsistente e incompletitud: LOGICAS LACANIANAS y Teoremas de la incompletitud de Gödel (Spanish)  

Jacques Lacan (Encyclopædia Britannica Online)

Usted puede darse cuenta de las relaciones internas entre ellos, y el hilo conductor es el título de este mismo post: “de la lógica a la ontología”.  

Probaré que no existe en absoluto tal construcción, que no se puede hacer desde las máquinas, y que no depende ni del hardware ni del software utilizado.   

Matizando la cuestión, el límite de la web semántica está dado no por las máquinas y/o sistemas biológicos que se pudieran usar, sino porque la lógica con que se intenta construir carece del uso del tiempo, ya que la lógica formal es puramente metonímica y carece de la metáfora, y eso es lo que marcan los teoremas de Gödel, la tautología final de toda construcción y /o lenguaje metonímico (matemático), que lleva a contradicciones.  

Esta lógica consistente es opuesta a la lógica inconsistente que hace uso del tiempo, propia del insconciente humano, pero el uso del tiempo está construido en base a la falta, no en torno a lo positivo sino en base a negaciones y ausencias, y eso es imposible de reflejar en una máquina porque la percepción de la falta necesita de la conciencia de sí mismo que se adquiere con la ausencia.   

El problema está en que pretendemos construir un sistema inteligente que sustituya nuestro pensamiento, al menos en las búsquedas de información, pero la particularidad de nuestro pensamiento humano es el uso del tiempo el que permite concluir, por eso no existe en la mente humana el problema de la parada o detención del cálculo, o lo que es lo mismo ausencia del momento de concluir.  

Así que todos los esfuerzos encaminados a la web semántica están destinados al fracaso a priori si lo que se pretende es prolongar nuestro pensamiento humano en las máquinas, ellas carecen de discurso metafórico, pues sólo son una construcción matemática, que siempre será tautológica y metonímica, ya que además carece del uso del tiempo que es lo que lleva al corte, la conclusión o la “parada”.  

Como demostración vale la del contraejemplo, o sea que si suponemos que es posible construir la web semántica, como un lenguaje con capacidades similares al lenguaje humano, que tiene el uso del tiempo, entonces si ese es un teorema general, con un solo contraejemplo se viene abajo, y el contraejemplo está dado en el caso particular de la máquina de Turing y el “problema de la parada”.  

Luego no se cumple la condición necesaria y suficiente del teorema, nos queda la condición necesaria que es que si un lenguaje tiene el uso del tiempo, carece de lógica formal, usa la lógica inconsistente y por lo tanto no tiene el problema de la parada”, esa es condición necesaria para la web semántica, pero no suficiente y por ello ninguna máquina, sea de Turing, computador o dispositivo aleatorio como un cuerpo negro en física, puede alcanzar el uso de un lenguaje que no sea el matemático con la paradoja de la parada, consecuencia del teorema de Gödel.

Jacques Lacan (Encyclopædia Britannica Online)

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Jacques Lacan (Encyclopædia Britannica Online) 

http://microscopia2007.blogspot.com/2007/10/logicas-lacanianas.html 

Lógica inconsistente e incompletitud: LOGICAS LACANIANAS y Teoremas de la incompletitud de Gödel

  

La revolución Gödeliana…del Otro que no existe.

Completud, incompletud, consistencia, inconsistencia, decidible e indecidible son conceptos de la metalógica que se refieren a ciertas características de los sistemas lógicos formales, más precisamente a los sistemas axiomáticos. Son conceptos que se atribuyen a K Gödel a partir de sus teoremas de principios del siglo anterior. Surgen en un contexto muy particular de las matemáticas en contraposición al ideal de David Hilbert que consideraba que en ese ámbito todo podría ser demostrable.

Kurt Gödel nació el 28 de abril de 1906 en Brünn, Moravia. Entró a formar parte del Círculo de Viena, siendo a partir de ese momento que comienza a elaborar sus teorías más importantes sobre la completitud de los sistemas formales a partir de dos publicaciones: su tesis doctoral escrita en 1929, y el teorema (Sobre proposiciones formalmente indecidibles en los Principia Mathematica y sistemas afines) publicado en 1931.En el año 1931, Gödel publicaba Sobre proposiciones…,artículo que ponía en cuestión el programa de D Hilbert, porque demostraba que no sólo el sistema de Russel y Whitehead tenía fisuras, sino que todo sistema axiomático los tendría.

Un sistema axiomático está compuesto por un conjunto de enunciados o fórmulas que se admiten sin demostración –axiomas- a partir de los cuales se obtienen todas las demás afirmaciones de la teoría llamadas teoremas. El conjunto de axiomas, más la definición de enunciado o fórmula del sistema (definición que precede al enunciado de los axiomas) y el conjunto de las reglas para la obtención de teoremas a partir de los axiomas (reglas de transformación) constituyen la base primitiva del sistema.

K. Gödel demostró que es imposible establecer la consistencia lógica interna de una amplia clase de sistemas deductivos, a menos que se adopten principios tan complejos de razonamiento que su consistencia interna quede tan sujeta a la duda como la de los propios sistemas, poniendo en juego la imposibilidad de demostrar ciertas proposiciones.

Consistencia, inconsistencia, completud e incompletud. ¿Qué es un sistema, qué significa que sea consistente, inconsistente, completo o incompleto, qué es una proposición, etcétera?
Un sistema es un conjunto de axiomas y reglas de inferencia, una proposición una afirmación que puede ser cierta o falsa. ¿Cuándo un sistema es completo? Cuando dentro de él puede determinarse el valor de verdad o falsedad de toda proposición.

La completud nos asegura que no hay ninguna verdad en nuestro sistema que nosotros no seamos capaces de encontrar Pero solo podremos estar seguros de poder alcanzar toda la verdad si nuestro sistema es completo.

En cambio es incompleto cuando contiene proposiciones sobre las que no podemos decidir su verdad o falsedad. Por otra parte, un sistema es coherente cuando no hay contradicciones de ningún tipo ni tiene ninguna paradoja; y es incoherente cuando nos encontramos con contradicciones y paradojas. Un sistema es consistente si está limpio de paradojas y contradicciones y completo si toda proposición puede ser demostrada o refutada entro de él. Gödel considera que si es consistente es incompleto y si es completo es inconsistente.

 

 

En ese sentido, la consistencia implica que no sea posible deducir, a partir del mismo sistema de axiomas, dos teoremas que sean contradictorios. Cuando se llega a una contradicción semántica, el sistema se muestra inconsistente.


El principio de inconsistencia entonces supone que el valor de verdad de un sistema no puede ser determinado a partir de un conjunto de axiomas sino solo desde un axioma exterior. Es decir que un sistema es inconsistente cuando no puede librarse de sus contradicciones semánticas internas.

  

Artículo original completo:

martes 2 de octubre de 2007

LOGICAS LACANIANAS

¿En qué consiste la Consistencia? (*)

-Usos de Lacan de los conceptos de consistencia, inconsistencia,
completud e incompletud-

Referencias
En su Curso anual Del witz que hay en el síntoma que dicta en la Asociación de Psicoanálisis de La Plata, Enrique Acuña introdujo las nociones de inconsistencia, consistencia, completud e incompletud a las que hace mención Lacan a lo largo de su enseñanza. En su última parte en el seminario 23 El sinthome, en relación al nudo borromeo define a lo imaginario por la consistencia, a lo simbólico por la inconsistencia en relación al equívoco significante, y a lo real por la ex -sistencia. Consistencia o inconsistencia del Otro, incompletud del Otro, consistencia lógica del objeto, consistencia de lo imaginario, son distintos enunciados a lo largo de la enseñanza de Lacan que van cobrando distintos sentidos.

 

 

José Ferrater Mora en su Diccionario de filosofía, destaca que el concepto de consistencia aparece en tres contextos diferentes: un uso en el que se describe la “real subsistencia en términos de consistencia”,un sentido metafísico en el que queda ligado al término esencia, por declararse que la esencia de algo es aquello en que este “algo” consiste – con cierta derivación hacia la noción de sustancia-, y por último un contexto lógico a partir de expresiones como prueba de consistencia por medio de la cual se prueba si un cálculo es o no consistente.La revolución Gödeliana…del Otro que no existe.
Completud, incompletud, consistencia, inconsistencia, decidible e indecidible son conceptos de la metalógica que se refieren a ciertas características de los sistemas lógicos formales, más precisamente a los sistemas axiomáticos. Son conceptos que se atribuyen a K Gödel a partir de sus teoremas de principios del siglo anterior. Surgen en un contexto muy particular de las matemáticas en contraposición al ideal de David Hilbert que consideraba que en ese ámbito todo podría ser demostrable
Kurt Gödel nació el 28 de abril de 1906 en Brünn, Moravia. Entró a formar parte del Círculo de Viena, siendo a partir de ese momento que comienza a elaborar sus teorías más importantes sobre la completitud de los sistemas formales a partir de dos publicaciones: su tesis doctoral escrita en 1929, y el teorema (Sobre proposiciones formalmente indecidibles en los Principia Mathematica y sistemas afines) publicado en 1931.En el año 1931, Gödel publicaba Sobre proposiciones…,artículo que ponía en cuestión el programa de D Hilbert, porque demostraba que no sólo el sistema de Russel y Whitehead tenía fisuras, sino que todo sistema axiomático los tendría.
Un sistema axiomático está compuesto por un conjunto de enunciados o fórmulas que se admiten sin demostración –axiomas- a partir de los cuales se obtienen todas las demás afirmaciones de la teoría llamadas teoremas. El conjunto de axiomas, más la definición de enunciado o fórmula del sistema (definición que precede al enunciado de los axiomas) y el conjunto de las reglas para la obtención de teoremas a partir de los axiomas (reglas de transformación) constituyen la base primitiva del sistema.
K. Gödel demostró que es imposible establecer la consistencia lógica interna de una amplia clase de sistemas deductivos, a menos que se adopten principios tan complejos de razonamiento que su consistencia interna quede tan sujeta a la duda como la de los propios sistemas, poniendo en juego la imposibilidad de demostrar ciertas proposicionesConsistencia, inconsistencia, completud e incompletud¿Qué es un sistema, qué significa que sea consistente, inconsistente, completo o incompleto, qué es una proposición, etcétera?
Un sistema es un conjunto de axiomas y reglas de inferencia, una proposición una afirmación que puede ser cierta o falsa. ¿Cuándo un sistema es completo? Cuando dentro de él puede determinarse el valor de verdad o falsedad de toda proposición
La completud nos asegura que no hay ninguna verdad en nuestro sistema que nosotros no seamos capaces de encontrar Pero solo podremos estar seguros de poder alcanzar toda la verdad si nuestro sistema es completo..En cambio es incompleto cuando contiene proposiciones sobre las que no podemos decidir su verdad o falsedad. Por otra parte, un sistema es coherente cuando no hay contradicciones de ningún tipo ni tiene ninguna paradoja; y es incoherente cuando nos encontramos con contradicciones y paradojas. Un sistema es consistente si está limpio de paradojas y contradicciones y completo si toda proposición puede ser demostrada o refutada entro de él. Gödel considera que si es consistente es incompleto y si es completo es inconsistente.
En ese sentido, la consistencia implica que no sea posible deducir, a partir del mismo sistema de axiomas, dos teoremas que sean contradictorios. Cuando se llega a una contradicción semántica, el sistema se muestra inconsistente.
El principio de inconsistencia entonces supone que el valor de verdad de un sistema no puede ser determinado a partir de un conjunto de axiomas sino solo desde un axioma exterior. Es decir que un sistema es inconsistente cuando no puede librarse de sus contradicciones semánticas internas.-Variaciones conceptuales: consistencia real, simbólica e imaginaria.
Consistencia del Otro, consistencia del objeto, consistencia de lo imaginario, inconsistencia e incompletitud del Otro…¿qué significado adquieren estos conceptos en estas afirmaciones de Lacan a lo largo de su enseñanza?
Al comienzo, más precisamente antes de la construcción del grafo del deseo, el Otro aparece sin barrar, esto es completo y consistente. Se trata de una consistencia simbólica en tanto adolece de contradicción semántica, y de una completud cuántica en cuanto ningún significante falta. Se trata de un Otro que la creencia neurótica construye.

La inconsistencia de este Otro – introducida por el equívoco significante que devela que no todo puede saberse- se revela con más fuerza en el seminario de La Angustia cuando construye el esquema de la doble causación del sujeto y del objeto a partir de la castración del Otro.
En el seminario De un Otro al otro, la expresión consistencia lógica aparece en relación a la nueva versión del objeto a que está construyendo ligada la versión del plus de gozar. Allí la consistencia no queda ligada a la versión de una lógica simbólica que supone un sistema axiomático libre de contradicción, sino más bien a la versión de una “real consistencia ligada a la esencia”. Real subsistencia en términos de consistencia ligada a la esencia, ya que en ella algo consiste. Esta versión se opone a la deriva de la cadena significante en la que no podemos encontrar ninguna consistencia definida en estos términos. Este objeto “sustancializado” viene a ocupar el lugar vacío del Otro, es decir que la consistencia del objeto toma su peso a partir de la inconsistencia el Otro. El objeto a en su consistencia, tapa la inconsistencia del Otro. Se opone así a la inconsistencia de la deriva de la cadena, la consistencia sustancial del objeto a.
En cuanto al registro imaginario, la completitud se pone en juego al comienzo en el estadio del espejo con relación a esa imagen que viene a velar la fragmentación real del organismo. Imagen completa que en tanto tal llena de júbilo al infans.

 

 

Por otro lado la consistencia en relación a lo imaginario Lacan la pone de manifiesto en el seminario El sinthome cuando la opone a la imposibilidad de lo real y a la inconsistencia semántica de lo simbólico introducida por el equívoco significante. Allí define a la consistencia imaginaria como “lo que mantiene junto”(1)

Enrique Acuña, en clase del 12 de septiembre de su curso anual Del witz que hay en el síntoma en la APLP, hizo mención a la versión del nudo borromeo que Lacan introduce en La Tercera según la cual, esta función de “lo que mantiene junto” la cumple el objeto a.
Es un “a” que da estabilidad. Planteó allí la necesidad de acompañar el trayecto que conduce a Lacan hacia la formulación del sinthome desde el nombre del padre, pasando por el objeto a teniendo como horizonte la pregunta ¿porqué Lacan sustituye en esa función de “lo que mantiene junto”, al objeto “a” por el sinthome?.
Se trata de los usos que Lacan hace de los conceptos extraídos de otras disciplinas- en este caso de la matemática, la lógica y la topología- para intentar decir cada vez de una manera nueva, eso que llamó su síntoma; lo real.
Marcelo Ale
Consultas:
Ferrater Mora Jose Diccionario de filosofía. Ariel, Barcelona. 1999.
Copi I Introducción a la
lógica. Manuales EUDEBA. Buenos Aires.1995.
Cohen, M y Ángel, E Introducción a la lógica y al método científico. Amorrortu. Buenos Aires. 1968. 2 vol.
WWW.Wikipedia.org/wiki teoremas.
Acuña Enrique Curso Anual APLP Del witz que hay en el síntoma. 2007
Lacan J La tercera en Intervenciones y textos 2 Manantial. Buenos Aires. 1988.
El seminario libro 23 El sinthome. Paidós. Buenos Aires. 2007.Notas:
(1) Lacan J. El Seminario 23 El sinthome, Página 63.

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Jacques Lacan (Encyclopædia Britannica Online)

The revolution … Another Gödel’s that does not exist!

Completeness, incompleteness, consistency, inconsistency, decidable and undecidable are concepts of meta logic which can be attributed to certain features of the formal logical systems, more precisely axiomatic systems. These are concepts that are attributed to K Gödel from their theorems from the beginning of the previous century. They emerge in a very particular context of mathematics as opposed to the ideal of David Hilbert who believed that everything in that area could be proof.

Kurt Gödel was born on April 28, 1906 in Brünn, Moravia. It became part of the Vienna Circle, and from that moment they begin to develop their most important theories on the completeness of the formal systems from two publications: his doctoral thesis written in 1929, and the theorem (formally on propositions undecidable in the Principia Mathematica and related systems) published in 1931.In 1931, Gödel published About propositions …, article that called into question the agenda D Hilbert, because not only showed that the system Russel and Whitehead had cracks, but the entire system would be axiomatic.

An axiomatic system consists of a set of formulas set forth or allowed without demonstration-axioms-from which all others are derived assertions theory called theorems. The set of axioms over the definition of phrasing or formula System (definition preceding statement of the axioms) and the set of rules for obtaining theorems from the axioms (transformation rules) are the basis of the primitive system.

K. Gödel proved that it is impossible to establish consistency internal logic of a broad class of deductive systems, unless it is taken early so complex reasoning that its internal consistency remains as subject to the doubt as to the systems themselves, putting at stake the impossibility proofing certain propositions. Consistency, inconsistency, completeness and incompleteness.What is a system, which means that it is consistently inconsistent, complete or incomplete, which is a proposition, etc.?
A system is a set of axioms and rules of inference, a claim that a proposition can be true or false. When a system is complete? Once inside it can be determined by the value of truth or falsity of any proposition
The completeness assures us that there is no truth in our system that we will not be able to find But we can only be sure of being able to reach the whole truth if our system is complete.Change is incomplete when it contains proposals on which we are unable to determine their truth or falsity. Moreover, a system is consistent when no contradictions of any kind nor does it have any paradox, and is inconsistent when we run into contradictions and paradoxes. A system is consistent if it is clean of paradoxes and contradictions and complete if any proposition can be proved or disproved sign him. Gödel believed that if it is consistent is incomplete and if it is completely inconsistent.In that sense, consistency means that it is not possible to deduce from the same set of axioms, two theorems which are contradictory. When it comes to contradiction semantics, the system is inconsistent.

The principle of inconsistency then assumed that the truth-value of a system can not be determined from a set of axioms, but only from a foreign axiom. That is a system that is inconsistent when it can not get rid of its internal contradictions semantic.

  

  

The article’s complete translation from Spanish Language is:

  

Tuesday October 2, 2007
LACAN’s LOGIC
 What is the consistency? (*)- Uses Lacan of the concepts of consistency, inconsistency, completeness and incompletenessReferences
At its annual course Witz is in the symptom that dictates in the Association of Psychoanalysis of La Plata, Enrique Acuña introduced the notions of inconsistency, consistency, and completeness and incompleteness to mention that Lacan over his teaching. In his last part in the seminar sinthome The 23, in relation to the Borromean knot defined by the imaginary consistency, as symbolic of the inconsistency in relation to significant misunderstanding, and what is real by the former existence. Consistency or inconsistency of the other incompletud Another, logical consistency of purpose, consistency of the imaginary, are different enunciated over the teaching of Lacan gaining different ways.

 

José Ferrater Mora in his Dictionary of Philosophy, stresses that the concept of consistency appears in three different contexts: a use which describes the “actual subsistence in terms of consistency,” a metaphysical sense in which the term is linked essence, by declaring that the essence of what something is that this “something” is – with some referral to the notion of substance, and finally a logical starting expressions as evidence of consistency by which it is tested whether a calculation is consistent or not.

The revolution … Another Gödel’s that does not exist.
Completeness, incompleteness, consistency, inconsistency, decidable and undecidable are concepts of meta logic which can be attributed to certain features of the formal logical systems, more precisely axiomatic systems. These are concepts that are attributed to K Gödel from their theorems from the beginning of the previous century. They emerge in a very particular context of mathematics as opposed to the ideal of David Hilbert who believed that everything in that area could be proof.
Kurt Gödel was born on April 28, 1906 in Brünn, Moravia. It became part of the Vienna Circle, and from that moment they begin to develop their most important theories on the completeness of the formal systems from two publications: his doctoral thesis written in 1929, and the theorem (formally on propositions undecidable in the Principia Mathematica and related systems) published in 1931.

In 1931, Gödel published About propositions …, article that called into question the agenda D Hilbert, because not only showed that the system Russel and Whitehead had cracks, but the entire system would be axiomatic.
An axiomatic system consists of a set of formulas set forth or allowed without demonstration-axioms-from which all others are derived assertions theory called theorems. The set of axioms over the definition of phrasing or formula System (definition preceding statement of the axioms) and the set of rules for obtaining theorems from the axioms (transformation rules) are the basis of the primitive system.
K. Gödel proved that it is impossible to establish consistency internal logic of a broad class of deductive systems, unless it is taken early so complex reasoning that its internal consistency remains as subject to the doubt as to the systems themselves, putting at stake the impossibility proofing certain propositions

Consistency, inconsistency, completeness and incompleteness

What is a system, which means that it is consistently inconsistent, complete or incomplete, which is a proposition, etc.?
A system is a set of axioms and rules of inference, a claim that a proposition can be true or false. When a system is complete? Once inside it can be determined by the value of truth or falsity of any proposition
The completeness assures us that there is no truth in our system that we will not be able to find But we can only be sure of being able to reach the whole truth if our system is complete.

. Change is incomplete when it contains proposals on which we are unable to determine their truth or falsity. Moreover, a system is consistent when no contradictions of any kind nor does it have any paradox, and is inconsistent when we run into contradictions and paradoxes. A system is consistent if it is clean of paradoxes and contradictions and complete if any proposition can be proved or disproved sign him. Gödel believed that if it is consistent is incomplete and if it is completely inconsistent.

In that sense, consistency means that it is not possible to deduce from the same set of axioms, two theorems which are contradictory. When it comes to a contradiction semantics, the system is inconsistent.
The principle of inconsistency then assumed that the truth-value of a system can not be determined from a set of axioms, but only from a foreign axiom. That is a system that is inconsistent when it can not get rid of its internal contradictions semantic.

– Variations concepts: consistency real and imaginary symbolic.
Another Consistency, consistency of purpose, consistency of the imaginary, inconsistency and incompleteness of the Other … what meaning acquire these concepts in these statements Lacan over their teaching?
Initially, more precisely before the construction of the graph of desire, without mark Another appears, that is complete and consistent. This is a symbolic while suffering from semantic contradiction, and a quantum completud as no significant fault. Another is a belief that the neurotic builds.

The inconsistency of this other – introduced by misleading significant that reveals that not everything can be known-is revealed with more force in the workshop of The Anxiety builds when the scheme of dual causation of the subject and the object from the castration of the Other .

The seminar From Another one to the other, the expression appears logical consistency in relation to the new version of the object being constructed linked to the release of more than enjoy. There consistency is not linked to the version of a symbolic logic that represents an axiomatic system free of contradiction, but rather the version of a “real consistency linked to the substance.” Real subsistence in terms of consistency linked to the substance, since it is something. This version is opposed to drift significant chain in which we were unable to find any consistency as defined in these terms. This object “substance” comes to take the place empty Another That is the consistency of the object takes its weight from the inconsistency the Other. The object in its consistency, cover the inconsistency of the Other. Opposes well to the inconsistency of the results from the chain, the consistency of the object substantial a.

As for the registration imaginary, comprehensiveness is at stake in the stadium at the beginning of the mirror with respect to that image that comes to ensuring actual fragmentation of the body. Image complete in itself full of joy to the baby.
On the other hand consistency in relation to the imaginary Lacan the shows at the seminar The sinthome when opposed to the inability of the real and the symbolic semantic inconsistency introduced by misleading significant. There defines consistency imaginary “which holds together” (1)

Enrique Acuña, class of 12 September from its current annual Witz is on the symptom in the APLP, referred to the version of the Borromean knot that Lacan introduces The Third whereby this function “which keeps together “meets the object a.
It is an “a” that gives stability. He raised hence the need to follow the path that leads to Lacan towards formulating the sinthome since his father’s name through the object to the horizon with the question why Lacan replaced in the role of “what holds together” in order “” by the sinthome?.

It is the uses that Lacan makes the concepts drawn from other disciplines-in this case of mathematics, logic and topology-to try to say every time a new way, that he called his symptoms, the real.

 

 

 An the original document is:

 

http://microscopia2007.blogspot.com/2007/10/logicas-lacanianas.html

 

************ EL PSICOANALISIS ENTRE LOS I N T E R s T I C I O S DE LA CULTURA * WWW.APLP.ORG.AR

martes 2 de octubre de 2007

LOGICAS LACANIANAS

¿En qué consiste la Consistencia? (*)

-Usos de Lacan de los conceptos de consistencia, inconsistencia,
completud e incompletud-

Referencias
En su Curso anual Del witz que hay en el síntoma que dicta en la Asociación de Psicoanálisis de La Plata, Enrique Acuña introdujo las nociones de inconsistencia, consistencia, completud e incompletud a las que hace mención Lacan a lo largo de su enseñanza. En su última parte en el seminario 23 El sinthome, en relación al nudo borromeo define a lo imaginario por la consistencia, a lo simbólico por la inconsistencia en relación al equívoco significante, y a lo real por la ex -sistencia. Consistencia o inconsistencia del Otro, incompletud del Otro, consistencia lógica del objeto, consistencia de lo imaginario, son distintos enunciados a lo largo de la enseñanza de Lacan que van cobrando distintos sentidos.

 

 

José Ferrater Mora en su Diccionario de filosofía, destaca que el concepto de consistencia aparece en tres contextos diferentes: un uso en el que se describe la “real subsistencia en términos de consistencia”,un sentido metafísico en el que queda ligado al término esencia, por declararse que la esencia de algo es aquello en que este “algo” consiste – con cierta derivación hacia la noción de sustancia-, y por último un contexto lógico a partir de expresiones como prueba de consistencia por medio de la cual se prueba si un cálculo es o no consistente.La revolución Gödeliana…del Otro que no existe.
Completud, incompletud, consistencia, inconsistencia, decidible e indecidible son conceptos de la metalógica que se refieren a ciertas características de los sistemas lógicos formales, más precisamente a los sistemas axiomáticos. Son conceptos que se atribuyen a K Gödel a partir de sus teoremas de principios del siglo anterior. Surgen en un contexto muy particular de las matemáticas en contraposición al ideal de David Hilbert que consideraba que en ese ámbito todo podría ser demostrable
Kurt Gödel nació el 28 de abril de 1906 en Brünn, Moravia. Entró a formar parte del Círculo de Viena, siendo a partir de ese momento que comienza a elaborar sus teorías más importantes sobre la completitud de los sistemas formales a partir de dos publicaciones: su tesis doctoral escrita en 1929, y el teorema (Sobre proposiciones formalmente indecidibles en los Principia Mathematica y sistemas afines) publicado en 1931.En el año 1931, Gödel publicaba Sobre proposiciones…,artículo que ponía en cuestión el programa de D Hilbert, porque demostraba que no sólo el sistema de Russel y Whitehead tenía fisuras, sino que todo sistema axiomático los tendría.
Un sistema axiomático está compuesto por un conjunto de enunciados o fórmulas que se admiten sin demostración –axiomas- a partir de los cuales se obtienen todas las demás afirmaciones de la teoría llamadas teoremas. El conjunto de axiomas, más la definición de enunciado o fórmula del sistema (definición que precede al enunciado de los axiomas) y el conjunto de las reglas para la obtención de teoremas a partir de los axiomas (reglas de transformación) constituyen la base primitiva del sistema.
K. Gödel demostró que es imposible establecer la consistencia lógica interna de una amplia clase de sistemas deductivos, a menos que se adopten principios tan complejos de razonamiento que su consistencia interna quede tan sujeta a la duda como la de los propios sistemas, poniendo en juego la imposibilidad de demostrar ciertas proposicionesConsistencia, inconsistencia, completud e incompletud
En ese sentido, la consistencia implica que no sea posible deducir, a partir del mismo sistema de axiomas, dos teoremas que sean contradictorios. Cuando se llega a una contradicción semántica, el sistema se muestra inconsistente.
El principio de inconsistencia entonces supone que el valor de verdad de un sistema no puede ser determinado a partir de un conjunto de axiomas sino solo desde un axioma exterior. Es decir que un sistema es inconsistente cuando no puede librarse de sus contradicciones semánticas internas.-Variaciones conceptuales: consistencia real, simbólica e imaginaria.
Consistencia del Otro, consistencia del objeto, consistencia de lo imaginario, inconsistencia e incompletitud del Otro…¿qué significado adquieren estos conceptos en estas afirmaciones de Lacan a lo largo de su enseñanza?
Al comienzo, más precisamente antes de la construcción del grafo del deseo, el Otro aparece sin barrar, esto es completo y consistente. Se trata de una consistencia simbólica en tanto adolece de contradicción semántica, y de una completud cuántica en cuanto ningún significante falta. Se trata de un Otro que la creencia neurótica construye.

La inconsistencia de este Otro – introducida por el equívoco significante que devela que no todo puede saberse- se revela con más fuerza en el seminario de La Angustia cuando construye el esquema de la doble causación del sujeto y del objeto a partir de la castración del Otro.
En el seminario De un Otro al otro, la expresión consistencia lógica aparece en relación a la nueva versión del objeto a que está construyendo ligada la versión del plus de gozar. Allí la consistencia no queda ligada a la versión de una lógica simbólica que supone un sistema axiomático libre de contradicción, sino más bien a la versión de una “real consistencia ligada a la esencia”. Real subsistencia en términos de consistencia ligada a la esencia, ya que en ella algo consiste. Esta versión se opone a la deriva de la cadena significante en la que no podemos encontrar ninguna consistencia definida en estos términos. Este objeto “sustancializado” viene a ocupar el lugar vacío del Otro, es decir que la consistencia del objeto toma su peso a partir de la inconsistencia el Otro. El objeto a en su consistencia, tapa la inconsistencia del Otro. Se opone así a la inconsistencia de la deriva de la cadena, la consistencia sustancial del objeto a.
En cuanto al registro imaginario, la completitud se pone en juego al comienzo en el estadio del espejo con relación a esa imagen que viene a velar la fragmentación real del organismo. Imagen completa que en tanto tal llena de júbilo al infans.

 

 

Por otro lado la consistencia en relación a lo imaginario Lacan la pone de manifiesto en el seminario El sinthome cuando la opone a la imposibilidad de lo real y a la inconsistencia semántica de lo simbólico introducida por el equívoco significante. Allí define a la consistencia imaginaria como “lo que mantiene junto”(1)

Enrique Acuña, en clase del 12 de septiembre de su curso anual Del witz que hay en el síntoma en la APLP, hizo mención a la versión del nudo borromeo que Lacan introduce en La Tercera según la cual, esta función de “lo que mantiene junto” la cumple el objeto a.
Es un “a” que da estabilidad. Planteó allí la necesidad de acompañar el trayecto que conduce a Lacan hacia la formulación del sinthome desde el nombre del padre, pasando por el objeto a teniendo como horizonte la pregunta ¿porqué Lacan sustituye en esa función de “lo que mantiene junto”, al objeto “a” por el sinthome?.
Se trata de los usos que Lacan hace de los conceptos extraídos de otras disciplinas- en este caso de la matemática, la lógica y la topología- para intentar decir cada vez de una manera nueva, eso que llamó su síntoma; lo real.
Marcelo Ale
Consultas:
Ferrater Mora Jose Diccionario de filosofía. Ariel, Barcelona. 1999.
Copi I Introducción a la
lógica. Manuales EUDEBA. Buenos Aires.1995.
Cohen, M y Ángel, E Introducción a la lógica y al método científico. Amorrortu. Buenos Aires. 1968. 2 vol.
WWW.Wikipedia.org/wiki teoremas.
Acuña Enrique Curso Anual APLP Del witz que hay en el síntoma. 2007
Lacan J La tercera en Intervenciones y textos 2 Manantial. Buenos Aires. 1988.
El seminario libro 23 El sinthome. Paidós. Buenos Aires. 2007.Notas:
(1) Lacan J. El Seminario 23 El sinthome, Página 63.

¿Qué es un sistema, qué significa que sea consistente, inconsistente, completo o incompleto, qué es una proposición, etcétera?
Un sistema es un conjunto de axiomas y reglas de inferencia, una proposición una afirmación que puede ser cierta o falsa. ¿Cuándo un sistema es completo? Cuando dentro de él puede determinarse el valor de verdad o falsedad de toda proposición
La completud nos asegura que no hay ninguna verdad en nuestro sistema que nosotros no seamos capaces de encontrar Pero solo podremos estar seguros de poder alcanzar toda la verdad si nuestro sistema es completo.

.En cambio es incompleto cuando contiene proposiciones sobre las que no podemos decidir su verdad o falsedad. Por otra parte, un sistema es coherente cuando no hay contradicciones de ningún tipo ni tiene ninguna paradoja; y es incoherente cuando nos encontramos con contradicciones y paradojas. Un sistema es consistente si está limpio de paradojas y contradicciones y completo si toda proposición puede ser demostrada o refutada entro de él. Gödel considera que si es consistente es incompleto y si es completo es inconsistente.

 

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Gödel’s incompleteness theorems

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In mathematical logic, Gödel’s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest.

The theorems are also of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert’s program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert’s second problem. Authors such as J. R. Lucas have argued that the theorems have implications in wider areas of philosophy and even cognitive science as well as preventing any complete Theory of Everything from being found in physics, but these claims are less generally accepted.

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[edit] First incompleteness theorem

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Here, “theory” refers to an infinite set of statements, some of which are taken as true without proof (these are called axioms), and others (the theorems) that are taken as true because they are implied by the axioms. “Provable in the theory” means “derivable from the axioms and primitive notions of the theory, using standard first-order logic“. A theory is “consistent” if it never proves a contradiction. “Can be constructed” means that some mechanical procedure exists which can construct the statement, given the axioms, primitives, and first order logic. “Elementary arithmetic” consists merely of addition and multiplication over the natural numbers. The resulting true but unprovable statement is often referred to as “the Gödel sentence” for the theory, although there are infinitely many other statements in the theory that share with the Gödel sentence the property of being true but not provable from the theory.

The hypothesis that the theory is computably enumerable means that it is possible in principle to write a computer program that (if allowed to run forever) would list all the theorems of the theory and no other statements. In fact, it is enough to enumerate the axioms in this manner since the theorems can then be effectively generated from them.

The first incompleteness theorem first appeared as “Theorem VI” in his 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I. In Gödel’s original notation, it states:

“The general result about the existence of undecidable propositions reads as follows:

“Theorem VI. For every ω-consistent recursive class κ of FORMULAS there are recursive CLASS SIGNS r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(κ) (where v is the FREE VARIABLE of r).2 (van Heijenoort translation and typsetting 1967:607. “Flg” is from “Folgerungsmenge = set of consequences” and “Gen” is from “Generalisation = generalization” (cf Meltzer and Braithwaite 1962, 1992 edition:33-34) )

Roughly speaking, the Gödel statement, G, asserts: “G cannot be proven true”. If G were able to be proven true under the theory’s axioms, then the theory would have a theorem, G, which contradicts itself, and thus the theory would be inconsistent. But if G were not provable, then it would be true (for G expresses this very fact) and thus the theory would be incomplete.

The argument just given is in ordinary English and thus not mathematically rigorous. In order to provide a rigorous proof, Gödel represented statements by numbers; then the theory, which is already about numbers, also pertains to statements, including its own. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence is a claim that there does not exist a natural number with a certain property. A number with that property would be a proof of inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to hypothesis. So, assuming the theory is consistent (as done in the theorem’s hypothesis) there is no such number, and the Gödel statement is true, but the theory cannot prove it. An important conceptual point is that we must assume that the theory is consistent in order to state that this statement is true.

[edit] Extensions of Gödel’s original result

Gödel demonstrated the incompleteness of the theory of Principia Mathematica, a particular theory of arithmetic, but a parallel demonstration could be given for any effective theory of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal theory.

Gödel’s original statement and proof of the incompleteness theorem requires the assumption that the theory is not just consistent but ω-consistent. A theory is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate P such that for every specific natural number n the theory proves ~P(n), and yet the theory also proves that there exists a natural number n such that P(n). That is, the theory says that a number with property P exists while denying that it has any specific value. The ω-consistency of a theory implies its consistency, but consistency does not imply ω-consistency. J. Barkley Rosser later strengthened the incompleteness theorem by finding a variation of the proof that does not require the theory to be ω-consistent, merely consistent. This is mostly of technical interest, since all true formal theories of arithmetic, that is, theories with only axioms that are true statements about natural numbers, are ω-consistent and thus Gödel’s theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, not ω-consistency, is now commonly known as Gödel’s incompleteness theorem.

[edit] Second incompleteness theorem

Gödel’s second incompleteness theorem can be stated as follows:

For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

(Proof of the “if” part:) If T is inconsistent then anything can be proved, including that T is consistent. (Proof of the “only if” part:) If T is consistent then T does not include the statement of its own consistency. This follows from the first theorem.

There is a technical subtlety involved in the second incompleteness theorem, namely how exactly are we to express the consistency of T in the language of T. There are many ways to do this, and not all of them lead to the same result. In particular, different formalizations of the claim that T is consistent may be inequivalent in T, and some may even be provable. For example, first order arithmetic (Peano arithmetic or PA for short) can prove that the largest consistent subset of PA is consistent. But since PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA “proves that it is consistent”. What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term “largest consistent subset of PA” is rather vague, but what is meant here is the largest consistent initial segment of the axioms of PA ordered according to some criteria, e.g. by “Gödel numbers”, the numbers encoding the axioms as per the scheme used by Gödel mentioned above).

In the case of Peano arithmetic or any familiar explicitly axiomatized theory T, it is possible to define the consistency “Con(T)” of T in terms of the non-existence of a number with a certain property, as follows: “there does not exist an integer coding a sequence of sentences, such that each sentence is either one of the (canonical) axioms of T, a logical axiom, or an immediate consequence of preceding sentences according to the rules of inference of first order logic, and such that the last sentence is a contradiction”. However, for arbitrary T there is no canonical choice for Con(T).

The formalization of Con(T) depends on two factors: formalizing the notion of a sentence being derivable from a set of sentences and formalizing the notion of being an axiom of T. Formalizing derivability can be done in canonical fashion, so given an arithmetical formula A(x) defining a set of axioms, we can canonically form the predicate ProvA(P) which expresses that P is provable from the set of axioms defined by A(x). Using this predicate we can express Con(T) as “not ProvA(‘P and not-P’)”. Solomon Feferman showed that Gödel’s second incompleteness theorem goes through when the formula A(x) is chosen so that it has the form “there exists a number n satisfying the decidable predicate P” for some P. In addition, ProvA(P) must satisfy the so-called HilbertBernays provability conditions:

1. If T proves P, then T proves ProvA(P)

2. T proves 1., i.e. T proves that if T proves P, then T proves ProvA(P)

3. T proves that if T proves that (P implies Q) then T proves that provability of P implies provability of Q

Gödel’s second incompleteness theorem also implies that a theory T1 satisfying the technical conditions outlined above can’t prove the consistency of any theory T2 which proves the consistency of T1. This is because then T1 can prove that if T2 proves the consistency of T1, then T1 is in fact consistent. For the claim that T1 is consistent has form “for all numbers n, n has the decidable property of not being a code for a proof of contradiction in T1“. If T1 were in fact inconsistent, then T2 would prove for some n that n is the code of a contradiction in T1. But if T2 also proved that T1 is consistent, i.e. there is no such n, it would itself be inconsistent. We can carry out this reasoning in T1 and conclude that if T2 is consistent, then T1 is consistent. Since by second incompleteness theorem, T1 does not prove its consistency, it can’t prove the consistency of T2 either.

This easy corollary of the second incompleteness theorem shows that there is no hope of proving e.g. the consistency of first order arithmetic using finitistic means provided we accept that finitistic means are correctly formalized in a theory the consistency of which is provable in PA. It’s generally accepted that the theory of primitive recursive arithmetic (PRA) is an accurate formalization of finitistic mathematics, and PRA is provably consistent in PA. Thus PRA can’t prove the consistency of PA. This is generally seen to show that Hilbert’s program, which is to use “ideal” mathematical principles to prove “real” (finitistic) mathematical statements by showing that the “ideal” principles are consistent by finitistically acceptable principles, can’t be carried out.

This corollary is actually what makes the second incompleteness theorem epistemically relevant. As Georg Kreisel remarked, it would actually provide no interesting information if a theory T proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of T in T would give us no clue as to whether T really is consistent; no doubts about T’s consistency would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory T in some theory T’ which is in some sense less doubtful than T itself, e.g. weaker than T. For most naturally occurring T and T’, such as T = Zermelo-Fraenkel set theory and T’ = primitive recursive arithmetic, the consistency of T’ is provable in T, and thus T’ can’t prove the consistency of T by the above corollary of the second incompleteness theorem.

The consistency of first-order arithmetic has been proved assuming that a certain ordinal called ε0 is wellfounded. See Gentzen’s consistency proof.

[edit] Original statement of Gödel’s Theorem XI

While contemporary usage calls it the “Second incompleteness Theorem”, in the original Gödel presented it as his “Theorem XI”. It is stated thus (in the following, “Section 2” is where his Theorem VI appears, and P is Gödel’s abbreviation for Peano Arithmetic ):

”The results of Section 2 have a surprising consequence concerning a consistency proof for the system P (and its extensions), which can be stated as follows:

”Theorem XI. Let κ be any recursive consistent63 class of FORMULAS; then the SENTENTIAL FORMULA stating that κ is consistent is not κ-PROVABLE; in particular, the consistency of P is not provable in P,64 provided P is consistent (in the opposite case, of course, every proposition is provable [in P])”. (Brackets in original added by Gödel “to help the reader”, translation and typography in van Heijenoort 1967:614)

63 “κ is consistent” (abbreviated by “Wid(κ)”) is defined as thus: Wid(κ)≡ (Ex)(Form(x) & ~Bewκ(x)).”

(Note: In the original “Bew” has a negation-“bar” written over it, indicated here by ~. “Wid” abbreviates “Widerspruchfreiheit = consistency”, “Form” abbreviates “Formel = formula”, “Bew” abbreviates “Beweisbar = provable” (translations from Meltzer and Braithwaite 1962, 1996 edition:33-34) )
64 This follows if we substitute the empty class of FORMULAS for κ.”

[edit] Meaning of Gödel’s theorems

Gödel’s theorems are theorems about first-order logic, and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs. (Automatic proof verification is closely related to automated theorem proving, though proving and checking the proof are usually different tasks.)

To be able to perform this process, we need to know what our axioms are. We could start with a finite set of axioms, such as in Euclidean geometry, or more generally we could allow an infinite list of axioms, with the requirement that we can mechanically check for any given statement whether it is an axiom from that set or not (an axiom schema). In computer science, this is known as having a recursive set of axioms. While an infinite list of axioms may sound strange, this is exactly what’s used in the usual axioms for the natural numbers, the Peano axioms: the inductive axiom is in fact an axiom schema — it states that if zero has any property and whenever any natural number has that property, its successor also has that property, then all natural numbers have that property — it does not specify which property and the only way to say in first-order logic that this is true of all definable properties is to have infinitely many statements, one for each property.

Gödel’s first incompleteness theorem shows that any formal system that includes enough of the theory of the natural numbers is incomplete: it contains statements that are neither provably true nor provably false. Or one might say, no formal system which aims to define the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. This has severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic.[1]

The existence of an incomplete system is in itself not particularly surprising. For example, if you take Euclidean geometry and you drop the parallel postulate, you get an incomplete system (in the sense that the system does not contain all the true statements about Euclidean space). A system can be incomplete simply because you haven’t discovered all the necessary axioms.

What Gödel showed is that in most cases, such as in number theory or real analysis, you can never create a complete and consistent finite list of axioms, or even an infinite list that can be produced by a computer program. Each time you add a statement as an axiom, there will always be other true statements that still cannot be proved as true, even with the new axiom. Furthermore if the system can prove that it is consistent, then it is inconsistent.

It is possible to have a complete and consistent list of axioms that cannot be produced by a computer program (that is, the list is not computably enumerable). For example, one might take all true statements about the natural numbers to be axioms (and no false statements). But then there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom or not.

Gödel’s theorem has another interpretation in the language of computer science. In first-order logic, theorems are computably enumerable: you can write a computer program that will eventually generate any valid proof. You can ask if they have the stronger property of being recursive: can you write a computer program to definitively determine if a statement is true or false? Gödel’s theorem says that in general you cannot.

Many logicians believe that Gödel’s incompleteness theorems struck a fatal blow to David Hilbert‘s program towards a universal mathematical formalism which was based on Principia Mathematica. The generally agreed-upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did.

[edit] Examples of undecidable statements

There are two distinct senses of the word “undecidable” in contemporary use. The first of these is the sense used in relation to Gödel’s theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system which proves for every question A in the problem either “the answer to A is yes” or “the answer to A is no”.

Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the “neither provable nor refutable” sense. The usage of “independent” is also ambiguous, however. Some use it to mean just “not provable”, leaving open whether an independent statement might be refuted.

Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called “absolutely undecidable” statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools.

One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by Max Dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent. This was shown to be the case in 1952.

The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.

In 1970, Soviet mathematician Yuri Matiyasevich showed that Hilbert’s Tenth Problem, posed in 1900 as a challenge to the next century of mathematicians, cannot be solved. Hilbert’s challenge sought an algorithm which finds all solutions of a Diophantine Equation. A Diophantine Equation is a more general case of Fermat’s Last Theorem; we seek the rational roots of a polynomial in any number of variables with integer coefficients. Since we have only one equation but n- variables, infinite solutions exist (and are easy to find) in the Complex Plane; the problem becomes difficult (impossible) by constraining solutions to rational values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine Equation to a recursively enumerable set and invoking Gödel’s Incompleteness Theorem.[2]

In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable, in the second sense of the term. This result was later generalized to Rice’s theorem.

In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory.

In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is undecidable in the axiomatization of arithmetic given by the Peano axioms but can be proven to be true in the larger system of second-order arithmetic.

Kruskal’s tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal’s tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.

Goodstein’s theorem is a statement about the Ramsey theory of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic.

Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin’s theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel’s theorem is related to the liar paradox, Chaitin’s result is related to Berry’s paradox.

Douglas Hofstadter gives a notable alternative proof of incompleteness, inspired by Gödel, in his book Gödel, Escher, Bach.

[edit] Limitations of Gödel’s theorems

The conclusions of Gödel’s theorems only hold for the formal systems that satisfy the necessary hypotheses (which have not been fully described in this article). Not all axiom systems satisfy these hypotheses, even when these systems have models that include the natural numbers as a subset. For example, there are first-order axiomatizations of Euclidean geometry and real closed fields that do not meet the hypotheses of Gödel’s theorems. The key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or develop basic properties of the natural numbers.

A second limitation is that Gödel’s theorems only apply to systems that are used as their own proof systems. For example, the consistency of the Peano arithmetic can be proved in set theory if set theory is consistent (however, one cannot prove that the latter is consistent in that framework). In 1936, Gerhard Gentzen proved the consistency of Peano arithmetic using a formal system which was more powerful in certain aspects than arithmetic, but less powerful than standard set theory.

[edit] Discussion and implications

The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system formal logic to define their principles. One can paraphrase the first theorem as saying, “we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods.”

On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical “truth” and “falsehood” are well-defined in an absolute sense, rather than relative to each formal system.

On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical “truth” and “falsehood” are well-defined in an absolute sense, rather than relative to each formal system.

The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

Therefore, in order to establish the consistency of a system S, one needs to use some other more powerful system T, but a proof in T is not completely convincing unless T’s consistency has already been established without using S.

At first, Gödel’s theorems seemed to leave some hope—it was thought that it might be possible to produce a general algorithm that indicates whether a given statement is undecidable or not, thus allowing mathematicians to bypass the undecidable statements altogether. However, the negative answer to the Entscheidungsproblem, obtained in 1936, showed that no such algorithm exists.

There are some who hold that a statement that is unprovable within a deductive system may be quite provable in a metalanguage. And what cannot be proven in that metalanguage can likely be proven in a meta-metalanguage, recursively, ad infinitum, in principle. By invoking such a system of typed metalanguages, along with an axiom of Reducibility — which by an inductive assumption applies to the entire stack of languages — one may, for all practical purposes, overcome the obstacle of incompleteness.

Note that Gödel’s theorems only apply to sufficiently strong axiomatic systems. “Sufficiently strong” means that the theory contains enough arithmetic to carry out the coding constructions needed for the proof of the first incompleteness theorem. Essentially, all that is required are some basic facts about addition and multiplication as formalized, e.g., in Robinson arithmetic Q. There are even weaker axiomatic systems that are consistent and complete, for instance Presburger arithmetic which proves every true first-order statement involving only addition.

The axiomatic system may consist of infinitely many axioms (as first-order Peano arithmetic does), but for Gödel’s theorem to apply, there has to be an effective algorithm which is able to check proofs for correctness. For instance, one might take the set of all first-order sentences which are true in the standard model of the natural numbers. This system is complete; Gödel’s theorem does not apply because there is no effective procedure that decides if a given sentence is an axiom. In fact, that this is so is a consequence of Gödel’s first incompleteness theorem.

Another example of a specification of a theory to which Gödel’s first theorem does not apply can be constructed as follows: order all possible statements about natural numbers first by length and then lexicographically, start with an axiomatic system initially equal to the Peano axioms, go through your list of statements one by one, and, if the current statement cannot be proven nor disproven from the current axiom system, add it to that system. This creates a system which is complete, consistent, and sufficiently powerful, but not computably enumerable.

Gödel himself only proved a technically slightly weaker version of the above theorems; the first proof for the versions stated above was given by J. Barkley Rosser in 1936.

In essence, the proof of the first theorem consists of constructing a statement p within a formal axiomatic system that can be given a meta-mathematical interpretation of:

p = “This statement cannot be proven in the given formal theory”

As such, it can be seen as a modern variant of the Liar paradox, although unlike the classical paradoxes it’s not really paradoxical.

If the axiomatic system is consistent, Gödel’s proof shows that p (and its negation) cannot be proven in the system. Therefore p is true (p claims to be not provable, and it is not provable) yet it cannot be formally proved in the system. If the axiomatic system is ω-consistent, then the negation of p cannot be proven either, and so p is undecidable. In a system which is not ω-consistent (but consistent), either we have the same situation, or we have a false statement which can be proven (namely, the negation of p).

Adding p to the axioms of the system would not solve the problem: there would be another Gödel sentence for the enlarged theory. Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are called essentially incomplete.

[edit] Minds and machines

Authors including J. R. Lucas have debated what, if anything, Gödel’s incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel’s incompleteness theorems would apply to it.

Hilary Putnam (1960) suggested that while Gödel’s theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are to believe that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing machine.

[edit] Postmodernism and continental philosophy

Appeals are sometimes made to the incompleteness theorems to support by analogy ideas which go beyond mathematics and logic. For instance, Régis Debray applies it to politics.[3] A number of authors have commented, mostly negatively, on such extensions and interpretations, including Torkel Franzen, Alan Sokal and Jean Bricmont, Ophelia Benson and Jeremy Stangroom. The last two quote[4] biographer Rebecca Goldstein[5] commenting on the disparity between Gödel’s avowed Platonism and the anti-realist uses to which his ideas are put by humanist intellectuals.

[edit] Theories of everything and physics

Stanley Jaki followed much later by Stephen Hawking and others argue that (an analogous argument to) Gödel’s theorem implies that even the most sophisticated formulation of physics will be incomplete, and that therefore there can never be an ultimate theory that can be formulated as a finite number of principles, known for certain as “final”. [6] [7]

[edit] Relationship with computability

As early as 1943, Kleene gave a proof of Godel’s incompleteness theorem using basic results of computability theory.[8] A basic result of computability shows that the halting problem is unsolvable: there is no computer program that can correctly determine, given a program P as input, whether P eventually halts when run with no input. Kleene showed that the existence of a complete effective theory of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. An exposition of this proof at the undergraduate level was given by Charlesworth (1980).[9]

By enumerating all possible proofs, it is possible to enumerate all the provable consequences of any effective first-order theory. This makes is possible to search for proofs of a certain form. Moreover, the method of arithmetization introduced by Gödel can be used to show that any sufficiently strong theory of arithmetic can represent the workings of computer programs. In particular, for each program P there is a formula Q such that Q expresses the idea that P halts when run with no input. The formula Q says, essentially, that there is a natural number that encodes the entire computation history of P and this history ends with P halting.

If, for every such formula Q, either Q or the negation of Q was a logical consequence of the axiom system, then it would be possible, by enumerating enough theorems, to determine which of these is the case. In particular, for each program P, the axiom system would either prove “P halts when run with no input,” or “P doesn’t halt when run with no input.”

Consistency assumptions imply that the axiom system is correct about these theorems. If the axioms prove that a program P doesn’t halt when the program P actually does halt, then the axiom system is inconsistent, because it is possible to use the complete computation history of P to make a proof that P does halt. This proof would just follow the computation of P step-by-step until P halts after a finite number of steps.

The mere consistency of the axiom system is not enough to obtain a contradiction, however, because a consistent axiom system could still prove the ω-inconsistent theorem that a program halts, when it actually doesn’t halt. The assumption of ω-consistency implies, however, that if the axiom system proves a program doesn’t halt then the program actually does not halt. Thus if the axiom system was consistent and ω-consistent, its proofs about which programs halt would correctly reflect reality. Thus it would be possible to effectively decide which programs halt by merely enumerating proofs in the system; this contradiction shows that no effective, consistent, ω-consistent formal theory of arithmetic that is strong enough to represent the workings of a computer can be complete.

[edit] Proof sketch for the first theorem

Throughout the proof we assume a formal system is fixed and satisfies the necessary hypotheses. The proof has three essential parts. The first part is to show that statements can be represented by natural numbers, known as Gödel numbers, and that properties of the statements can be detected by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that a statement is provable in the system. The second part of the proof is to construct a particular statement that, essentially, says that it is unprovable. The third part of the proof is to analyze this statement to show that is neither provable nor disprovable in the system.

[edit] Arithmetization of syntax

The main problem in fleshing out the above mentioned proof idea is the following: in order to construct a statement p that is equivalent to “p cannot be proved”, p would have to somehow contain a reference to p, which could easily give rise to an infinite regress. Gödel’s ingenious trick, which was later used by Alan Turing in his work on the Entscheidungsproblem, is to represent statements as numbers, which is often called the arithmetization of syntax.

To begin with, every formula or statement that can be formulated in our system gets a unique number, called its Gödel number. This is done in such a way that it is easy to mechanically convert back and forth between formulas and Gödel numbers. It is similar, for example, to the way English sentences are encoded as sequences (or “strings”) of numbers using ASCII: such a sequence is considered as a single (if potentially very large) number. Because our system is strong enough to reason about numbers, it is now also possible to reason about formulas within the system.

A formula F(x) that contains exactly one free variable x is called a statement form or class-sign. As soon as x is replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number n, F(n) is true if and only if it can be proven (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as “2*3=6”.

Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form F(x) can be assigned with a Gödel number which we will denote by G(F). The choice of the free variable used in the form F(x) is not relevant to the assignment of the Gödel number G(F).

Now comes the trick: The notion of provability itself can also be encoded by Gödel numbers, in the following way. Since a proof is a list of statements which obey certain rules, we can define the Gödel number of a proof. Now, for every statement p, we may ask whether a number x is the Gödel number of its proof. The relation between the Gödel number of p and x, the Gödel number of its proof, is an arithmetical relation between two numbers. Therefore there is a statement form Bew(x) that uses this arithmetical relation to state that a Gödel number of a proof of x exists:

Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y).

The name Bew is short for beweisbar, the German word for “provable”. An important feature of Bew is that if a statement p is provable in the system then Bew(G(p)) is also provable. This is because any proof of p would have a corresponding Gödel number, the existence of which causes Bew(G(p)) to be satisfied.

[edit] Diagonalization

The next step in the proof is to obtain a statement that says it is unprovable. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement form F there is a statement p such that the system proves

pF(G(p)).

We obtain p by letting F be the negation of Bew(x); thus p roughly states that its own Gödel number is the Gödel number of an unprovable formula.

The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of p itself. This is similar to the following sentence in English:

“, when preceded by itself in quotes, is unprovable.”, when preceded by itself in quotes, is unprovable.

This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence asserts its own unprovability. The proof of the diagonal lemma employs a similar method.

[edit] Proof of independence

We will now assume that our axiomatic system is ω-consistent. We let p be the statement obtained in the previous section.

If p were provable, then Bew(G(p)) would be provable, as argued above. But p asserts the negation of Bew(G(p)). Thus our system would be inconsistent, proving both a statement and its negation. This contradiction shows that p cannot be provable.

If the negation of p were provable, then Bew(G(p)) would be provable (because p was constructed to be equivalent to the negation of Bew(G(p))). However, for each specific number x, x cannot be the Gödel number of the proof of p, because p is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of p), but on the other hand, for every specific number x, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of p is not provable.

So the statement p is undecidable: it can neither be proved nor disproved within our system. ∎

It should be noted that p is not provable (and thus true) in every consistent system. The assumption of ω-consistency is only required for the negation of p to be not provable. Thus:

  • In an ω-consistent formal system, we may prove neither p nor its negation, and so p is undecidable.
  • In a consistent formal system we may either have the same situation, or we may prove the negation of p; In the later case, we have a statement (“not p“) which is false but provable.

Note that if one tries to “add the missing axioms” in order to avoid the undecidability of the system, then one has to add either p or “not p” as axioms. But then the definition of “being a Gödel number of a proof” of a statement changes. which means that the statement form Bew(x) is now different. Thus when we apply the diagonal lemma to this new form Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new system if it is ω-consistent.

Rosser (1936) showed, by employing a Gödel sentence more complicated than p, that ordinary consistency sufficed for this proof.

[edit] Boolos’s short proof

George Boolos (1998) vastly simplified the proof of the First Theorem, if one agrees that that theorem is equivalent to:

“There is no algorithm M whose output contains all true sentences of arithmetic and no false ones.”

“Arithmetic” refers to Peano or Robinson arithmetic, but the proof invokes no specifics of either. It is tacitly assumed that these systems allow ‘<‘ and ‘×’ to have their usual meanings (these are also the only defined arithmetical notions the proof requires). The Gödel sentence draws on Berry’s paradox, except that “fewer than n symbols of the language of arithmetic” replace “fewer than n natural language syllables.” Boolos proves the theorem in about two pages, employing the language of first order logic but invoking no facts about the connectives or quantifiers. The domain is the natural numbers, but the proof is innocent of infinity in any form.

Let [n] abbreviate (the natural number) n successive applications of the successor function, starting from 0. Boolos then defines several related predicates, starting with Cxz, which comes out true iff an arithmetic formula containing z symbols “names” (see below) the number x. The construction of C is only sketched. This sketch assumes that every formula has a Gödel number; this is the only mention of Gödel numbering in the entire proof. The other predicates are:

Bxy ↔ ∃z(z<yCxz),
Axy ↔ ¬Bxy ∧ ∀a(a<xBay),
Fx ↔ ∃y((y=[10]×[k]) ∧ Axy). k = the number of symbols appearing in Axy.

Fx “names” n if the output of M includes the sentence ∀x(Fx ↔(x=[n])). Thus Berry’s paradox is formalized. The balance of the proof, requiring but 12 lines of text, shows that this sentence is true in a semantic sense, but no algorithm M will identify it as true. Thus arithmetic truth outruns proof. QED.

The proof is intuitionistically valid, and requires but two existential quantifiers. The proof nowhere mentions recursive functions or any facts from number theory; Boolos even claims that the proof dispenses with diagonalization. For more on this proof, see Berry’s paradox.

[edit] Proof sketch for the second theorem

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate for provability. Once this is done, the second incompleteness theorem essentially follows by formalizing the entire proof of the first incompleteness theorem within the system itself.

Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be proven from within the system itself. We have seen above that if the system is consistent, then p is not provable. The proof of this implication can be formalized within the system, and therefore the statement “p is not provable”, or “not P(p)” can be proven in the system.

But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can be proven in the system. This contradiction shows that the system must be inconsistent.

[edit] See also

[edit] Footnotes

1 The word “true” here is being used disquotationally; that is, the statement “GT is true” means the same thing as GT itself. Thus a formalist might reinterpret the claim

for every theory T satisfying the hypotheses, if T is consistent, then GT is true

to mean

for every theory T satisfying the hypotheses, it is a theorem of Peano Arithmetic that Con(T)→GT

where Con(T) is the natural formalization of the claim “T is consistent”, and GT is the Gödel sentence for T.

2 Here Flg(κ) represents the theory generated by κ and “v Gen r” is a particular formula in the language of arithmetic.

[edit] References

[edit] In-text references

  1. ^ Geoffrey Hellman, How to Gödel a Frege-Russell: Gödel’s Incompleteness Theorems and Logicism. Noûs, Vol. 15, No. 4, Special Issue on Philosophy of Mathematics. (Nov., 1981), pp. 451-468.
  2. ^ Enumerable sets are Diophantine, Yuri Matiyasevich (1970). Doklady Akademii Nauk SSSR, 279-82. 
  3. ^ “The secret takes the form of a logical law, an extension of Gödel’s theorem: There can be no organised system without closure and no system can be closed by elements internal to that system alone“.Debray, R. Critique of Political Reason, quoted in Sokal and Bricmont’s Fashionable Nonsense.
  4. ^ In their Why Truth Matters
  5. ^ The Proof and paradox of Kurt Gödel
  6. ^ Stanley Jaki“A Late Awakening to Gödel in Physics”
  7. ^ Stephen Hawking “Gödel and the end of physics”
  8. ^ Kleene 1943, Theorem VIII.
  9. ^ A more rigorous proof-sketch can be found on pages 354 and 371 in John Hopcroft and Jeffrey Ullman 1979, Introduction to Automata theory, Addison-Wesley, ISBN 0-201-02988-X. More insight into the notion of “proofs as strings of symbols on Turing machines” can be found in p.221-226ff of Marvin Minsky 1967, Computation: Finite and Infinite Machines, Prentice-Hall, NJ, no ISBN. Minsky’s argument relies on the question that if one were to start with a “theorem” (i.e. a symbol-string that represents the last line of possible proof) and a machine to generate well-formed proofs as strings of symbols, will the machine ever match the theorem to a proof and then halt? It may make a match, but then again it may not, and it must continue forever searching. In general, this halting problem is undecidable. See more at tag system and Post correspondence problem.

[edit] Articles by Gödel

[edit] Translations, during his lifetime, of Gödel’s paper into English

None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize “those metamathematical notions that had been defined in their usual sense before . . .”(van Heijenoort 1967:595). Three translations exist. Of the first John Dawson states that: “The Meltzer translation was seriously deficient and received a devastating review in the Journal of Symbolic Logic; ”Gödel also complained about Braithwaite’s commentary (Dawson 1997:216). “Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis’s anthology The Undecidable . . . he found the translation “not quite so good” as he had expected . . . [but due to time constraints he] agreed to its publication” (ibid). (In a footnote Dawson states that “he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints” (ibid)). Dawson states that “The translation that Gödel favored was that by Jean van Heijenoort”(ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser “during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934” (cf commentary by Davis 1965:39 and beginning on p. 41); this version is titled “On Undecidable Propositions of Formal Mathematical Systems”. In their order of publication:

  • B. Meltzer (translation) and R. B. Braithwaite (Introduction), 1962. On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover Publications, New York (Dover edition 1992), ISBN 0-486-66980-7 (pbk.) This contains a useful translation of Gödel’s German abbreviations on pp.33-34. As noted above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all its suspect content by
  • Stephen Hawking editor, 2005. God Created the Integers: The Mathematical Breakthroughs That Changed History, Running Press, Philadelphia, ISBN-10: 0-7624-1922-9. Gödel’s paper appears starting on p. 1097, with Hawking’s commentary starting on p. 1089.
  • Martin Davis editor, 1965. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions, Raven Press, New York, no ISBN. Godel’s paper begins on page 5, preceded by one page of commentary.
  • Jean van Heijenoort editor, 1967, 3rd edition 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1979-1931, Harvard University Press, Cambridge Mass., ISBN 0-674-32449-8 (pbk). van Heijenoort did the translation. He states that “Professor Gödel approved the translation, which in many places was accommodated to his wishes.”(p. 595). Gödel’s paper begins on p. 595; van Heijenoort’s commentary begins on p. 592.
  • Martin Davis editor, 1965, ibid. “On Undecidable Propositions of Formal Mathematical Systems.” A copy with Gödel’s corrections of errata and Gödel’s added notes begins on page 41, preceded by two pages of Davis’s commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes.

[edit] Articles by others

  • George Boolos, 1998, “A New Proof of the Gödel Incompleteness Theorem” in Boolos, G., Logic, Logic, and Logic. Harvard Univ. Press.
  • Arthur Charlesworth, 1980, “A Proof of Godel’s Theorem in Terms of Computer Programs,” Mathematics Magazine, v. 54 n. 3, pp. 109-121. JStor
  • David Hilbert, 1900, “Mathematical Problems.” English translation of a lecture delivered before the International Congress of Mathematicians at Paris, containing Hilbert’s statement of his Second Problem.
  • Putnam, Hilary, 1960, Minds and Machines in Sidney Hook, ed., Dimensions of Mind: A Symposium. New York University Press. Reprinted in Anderson, A. R., ed., 1964. Minds and Machines. Prentice-Hall: 77.
  • Stephen Kleene, 1943, “Recursive predicates and quantifiers,” reprinted from Transactions of the American Mathemaical Society, v. 53 n. 1, pp. 41–73 in Martin Davis 1965, The Undecidable (loc. cit.) pp. 255–287.
  • John Barkley Rosser, 1936, “Extensions of some theorems of Gödel and Church,” reprinted from the Journal of Symbolic Logic vol. 1 (1936) pp. 87-91, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 230-235.
  • John Barkley Rosser, 1939, “An Informal Exposition of proofs of Gödel’s Theorem and Church’s Theorem”, Reprinted from the Journal of Symbolic Logic, vol. 4 (1939) pp. 53-60, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 223-230
  • Jean van Heijenoort, 1963. “Gödel’s Theorem” in Edwards, Paul, ed., Encyclopedia of Philosophy, Vol. 3. Macmillan: 348-57.

[edit] Books about the theorems

[edit] Miscellaneous References

  • John W. Dawson, Jr., 1997. Logical Dilemmas: The Life and Work of Kurt Gödel, A.K. Peters, Wellesley Mass, ISBN 1-56881-256-6.
  • Goldstein, Rebecca, 2005, Incompleteness – The Proof & Paradox of Kurt Godel.

[edit] External links

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