Re: Question about first-order arithmetic

Peter_Smith wrote:
I'm reminded of Feferman's observation, re Friedman's results, that it
overshoots to say that this or that arithmetical proposition has to be
proved appealing to heavy-duty set theoretic assumptions if -- when the
wraps are off -- it is only the consistency or 1-consistency of the
relevant set theory that is really needed.

Feferman is right in purely technical sense (of course). However, I'm
not sure what is the relevance of this technical observation; for it
seems there is no reason to accept the 1-consistency of ZFC + there is
a Mahlo cardinal or ZFC + there exists a ridiculous amount of Woodin
cardinals or whatever, that would not also lead one to accept the
soundness of the theory. This is a purely empirical observation; we can
easily *imagine* there being such a reason, e.g. if we had an ordinal
analysis of ZFC + whatever and could convince ourselves of primitive
recursive transfinite induction on the ordinal notation system, while
holding that set theoretical assertions are "meaningless" in
themselves. But such fantasies aside, as things stand today it seems
somewhat pointless to note that only 1-consistency of some set
theoretical assumption is needed to prove this or that, since it's
highly unlikely that if mathematicians come to accept e.g. Friedman's
theorems they will accept them because they find the set theoretical
reasoning convincing, not because they come to accept 1-consistency of
the relevant set theoretical principles.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logic-Philosophicus

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Relevant Pages

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