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A consistency proof is a mathematical proof that a logical system is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert’s program. Hilbert’s program fell to Gödel’s insight, as expressed in his two incompleteness theorems, that sufficiently strong proof theories cannot prove their own consistency.
Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.
 Consistency and completeness
The fundamental results relating consistency and completeness were proven by Kurt Gödel:
- Gödel’s completeness theorem shows that any consistent first-order theory is complete with respect to a maximal consistent set of formulae which are generated by means of a proof search algorithm.
- Gödel’s incompleteness theorems show that theories capable of expressing their own provability relation and of carrying out a diagonal argument are capable of proving their own consistency only if they are inconsistent. Such theories, if consistent, are known as essentially incomplete theories.
By applying these ideas, we see that we can find first-order theories of the following four kinds:
- Inconsistent theories, which have no models;
- Theories which cannot talk about their own provability relation, such as Tarski’s axiomatisation of point and line geometry, and Presburger arithmetic. Since these theories are satisfactorily described by the model we obtain from the completeness theorem, such systems are complete;
- Theories which can talk about their own consistency, and which include the negation of the sentence asserting their own consistency. Such theories are complete with respect to the model one obtains from the completeness theorem, but contain as a theorem the derivability of a contradiction, in contradiction to the fact that they are consistent;
- Essentially incomplete theories.
In addition, it has recently been discovered that there is a fifth class of theory, the self-verifying theories, which are strong enough to talk about their own provability relation, but are too weak to carry out Gödelian diagonalisation, and so which can consistently prove their own consistency. However as with any theory, a theory proving its own consistency provides us with no interesting information, since inconsistent theories also prove their own consistency.
Φ is said to be absolutely consistent or Post consistent iff at least one formula of Φ is not a theorem of Φ.
Φ is said to be maximally consistent if and only if for every formula φ, if Con then .
Φ is said to contain witnesses if and only if for every formula of the form there exists a term t such that . See First-order logic.
 Basic Results
1. The following are equivalent:
(b) For all
2. Every satisfiable set of formulas is consistent, where a set of formulas Φ is satisfiable if and only if there exists a model such that .
3. For all Φ and φ:
(a) if not , then Con;
(b) if Con Φ and , then Con;
(c) if Con Φ, then Con or Con.
4. Let Φ be a maximally consistent set of formulas and contain witnesses. For all φ and ψ:
(a) if , then ,
(b) either or ,
(c) if and only if or ,
(d) if and , then ,
(e) if and only if there is a term t such that .
 Henkin’s Theorem
Let Φ be a maximally consistent set of formulas containing witnesses.
Define a binary relation on the set of S-terms if and only if ; and let denote the equivalence class of terms containing ; and let where is the set of terms based on the symbol set .
Define the S-structure over the term-structure corresponding to Φ by:
(1) For n-ary , if and only if ,
(2) For n-ary , ,
(3) For , .
Let be the term interpretation associated with Φ, where .
 Sketch of Proof
There are several things to verify. First, that is an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of class representatives. Finally, can be verified by induction on formulas.
 See also
- Hilbert’s second problem
- Hilbert’s program
- Hilbert’s problems
- Matiyasevich’s theorem
- Emil Post (1920)
H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic