is arithmetic finitely consistent?

i have also asked this question on MSE - https://math.stackexchange.com/questions/4863426/is-arithmetic-finitely-consistent

and reddit - https://www.reddit.com/r/learnmath/comments/1arhu7o/is_arithmetic_finitely_consistent/

no reply , as of the moment i'm writing this post.

Let's take PA1 for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest consistent subset of the theory is PA1 itself.

But PA1 has infinite axioms, so can PA1 prove atleast for a given finite set of axioms ( of PA1 ) that they are consistent, specifically that no contradictional proof exists which uses only those axioms ?

Or any formal theory of arithmetic for that matter, can it prove for a finite subset of its own axioms that they are consistent ?

posted 9 months ago

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Jason‭

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