Gentzen's consistency proof

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Gentzen's consistency proof is a result of proof theory in mathematical logic. It "reduces" the consistency of a simplified part of mathematics, not to something that could be proved in that same simplified part of mathematics (which would contradict the basic results of Kurt Gödel), but rather to a simpler logical principle.

Gentzen's theorem[edit]

In 1936 Gerhard Gentzen proved the consistency of first-order arithmetic using combinatorial methods. Gentzen's proof shows much more than merely that first-order arithmetic is consistent. Gentzen showed that the consistency of first-order arithmetic is provable, over the base theory of primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0. Informally, this additional principle means that there is a well-ordering on the set of finite rooted trees.

The principle of quantifier-free transfinite induction up to ε0 says that for any formula A(x) with no bound variables transfinite induction up to ε0 holds. ε0 is the first ordinal \alpha, such that \omega^\alpha = \alpha, i.e. the limit of the sequence:

\omega,\ \omega^\omega,\ \omega^{\omega^\omega},\ \ldots

To express ordinals in the language of arithmetic an ordinal notation is needed, i.e. a way to assign natural numbers to ordinals less than ε0. This can be done in various ways, one example provided by Cantor's normal form theorem. That transfinite induction holds for a formula A(x) means that A does not define an infinite descending sequence of ordinals smaller than ε0 (in which case ε0 would not be well-ordered). Gentzen assigned ordinals smaller than ε0 to proofs in first-order arithmetic and showed that if there is a proof of contradiction, then there is an infinite descending sequence of ordinals < ε0 produced by a primitive recursive operation on proofs corresponding to a quantifier-free formula.

Relation to Gödel's theorem[edit]

Gentzen's proof also highlights one commonly missed aspect of Gödel's second incompleteness theorem. It is sometimes claimed that the consistency of a theory can only be proved in a stronger theory. The theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order arithmetic but is not stronger than first-order arithmetic. For example, it does not prove ordinary mathematical induction for all formulae, while first-order arithmetic does (it has this as an axiom schema). The resulting theory is not weaker than first-order arithmetic either, since it can prove a number-theoretical fact - the consistency of first-order arithmetic - that first-order arithmetic cannot. The two theories are simply incomparable.

Gentzen's proof is the first example of what is called proof-theoretical ordinal analysis. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can be proven to be well-ordered, or equivalently for how large a (constructive) ordinal can transfinite induction be proven. A constructive ordinal is the order type of a recursive well-ordering of natural numbers.

Laurence Kirby and Jeff Paris proved in 1982 that Goodstein's theorem cannot be proven in Peano arithmetic based on Gentzen's theorem.