Re: Peano arithmetic and transfinite induction

On Aug 25, 9:53 pm, djr...@xxxxxxxxxx wrote:
Famously, Godel's theorems proved that one could not prove the
consistency of Peano arithmetic (PA) within PA. As I understand it,
there is a diophantine diophantine equation which (iff PA is
consistent) has no solutions, but PA cannot prove this fact.

Gentzen proved the consistency of PA by assuming a principle of
transfinite induction. However, I often find quotes such as: "It is
evidently necessary to assume the validity of something like
transfinite induction to prove the consistency of arithmetic". Surely
just because con(PA) cannot be proved in PA and can be proved in PA
+transfinite induction, does not mean that PA+transfinite induction is
*precisely* what is needed to prove con(PA)? I would have thought that
all it tells us, is that PA+transfinite induction contains what is
required to prove con(PA). Is there some good reason for believing
this quoted statement of which I am not aware?

What I was trying to say really, surely there is no reason in
principle that we could not, using arithmetical methods not accounted
for in PA, find a proof of con(PA) which Fermat (transfinite
principles being alien to him) would recognise?

.

Relevant Pages

  • Re: Peano arithmetic and transfinite induction
    ... there is a diophantine diophantine equation which (iff PA is ... evidently necessary to assume the validity of something like ... transfinite induction to prove the consistency of arithmetic". ...
    (sci.logic)
  • Peano arithmetic and transfinite induction
    ... there is a diophantine diophantine equation which (iff PA is ... I often find quotes such as: ... evidently necessary to assume the validity of something like ... transfinite induction to prove the consistency of arithmetic". ...
    (sci.logic)
  • Re: incompleteness and inconsistency
    ... Even if the consistency of transfinite induction can be reasonably ... argument were the only reason we have for thinking that PA is ... relative consistency proof with respect to ZF or ZFC? ...
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