Consistency Peano axioms

when peano axioms first proposed, bertrand russell , others agreed these axioms implicitly defined mean natural number . henri poincaré more cautious, saying defined natural numbers if consistent; if there proof starts these axioms , derives contradiction such 0 = 1, axioms inconsistent, , don t define anything. in 1900, david hilbert posed problem of proving consistency using finitistic methods second of twenty-three problems. in 1931, kurt gödel proved second incompleteness theorem, shows such consistency proof cannot formalized within peano arithmetic itself.

although claimed gödel s theorem rules out possibility of finitistic consistency proof peano arithmetic, depends on 1 means finitistic proof. gödel himself pointed out possibility of giving finitistic consistency proof of peano arithmetic or stronger systems using finitistic methods not formalizable in peano arithmetic, , in 1958, gödel published method proving consistency of arithmetic using type theory. in 1936, gerhard gentzen gave proof of consistency of peano s axioms, using transfinite induction ordinal called ε0. gentzen explained: aim of present paper prove consistency of elementary number theory or, rather, reduce question of consistency fundamental principles . gentzen s proof arguably finitistic, since transfinite ordinal ε0 can encoded in terms of finite objects (for example, turing machine describing suitable order on integers, or more abstractly consisting of finite trees, suitably linearly ordered). whether or not gentzen s proof meets requirements hilbert envisioned unclear: there no accepted definition of meant finitistic proof, , hilbert himself never gave precise definition.

the vast majority of contemporary mathematicians believe peano s axioms consistent, relying either on intuition or acceptance of consistency proof such gentzen s proof. small number of philosophers , mathematicians, of whom advocate ultrafinitism, reject peano s axioms because accepting axioms amounts accepting infinite collection of natural numbers. in particular, addition (including successor function) , multiplication assumed total. curiously, there self-verifying theories similar pa have subtraction , division instead of addition , multiplication, axiomatized in such way avoid proving sentences correspond totality of addition , multiplication, still able prove true




Π

1




{\displaystyle \pi _{1}}

theorems of pa, , yet can extended consistent theory proves own consistency (stated non-existence of hilbert-style proof of 0=1 ).