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 weaker base theory of primitive recursive arithmetic
with the additional principle of quantifier free transfinite induction
up to the ordinal
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 , such that , i.e. the limit of the sequence:
To express ordinals in the language of arithmetic a 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.
Gentzen's proof also highlights one commonly missed aspect of Gödel's second incompleteness theorem. It's 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 doesn't 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 leads to Gentzen's theorem, i.e. it can substitute for induction up to ε0.
- G. Gentzen, 1936. 'Die Widerspruchfreiheit der reinen Zahlentheorie'. Mathematische Annalen, 112:493–565. Translated as 'The consistency of arithmetic', in (M. E. Szabo 1969).
- G. Gentzen, 1938. 'Neue Fassung des Widerspruchsfreiheitsbeweises fuer die reine Zahlentheorie'. Translated as 'New version of the consistency proof for elementary number theory', in (M. E. Szabo 1969).
- K. Gödel, 1938. Lecture at Zilsel’s, In Feferman et al. Kurt Gödel: Collected Works, Vol III, pp. 87–113.
- M. E. Szabo (ed.), 1969. 'The collected works of Gerhard Gentzen. North-Holland, Amsterdam.
- W. W. Tait, 2005. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counterexample interpretation The Bulletin of Symbolic Logic 11(2):225-238.
- Kirby, L. and Paris, J., Accessible independence results for Peano arithmetic, Bull. London. Math. Soc., 14 (1982), 285-93.