Re: Torkel Franzen on truth

On Dec 15, 8:33 am, Peter_Smith <ps...@xxxxxxxxx> wrote:
On 15 Dec, 04:27, Newberry <newberr...@xxxxxxxxx> wrote:

I am amazed that people claim that when a system proves its own
consistency the proof does not have any cogency but when you prove
it is in a stronger system it does.

Tertium datur, of course. You may be able to prove the consistency of
one theory in another theory which is neither strictly weaker nor
strictly stronger than the other. This is what happens with Gentzen's
proof of the consistency of first-order PA using PRA + transfinite
induction up to epsilon_0 for quantifier free formulae.

So noted. I also note that most people do not find Gentzen's proof
very convincing. And I still do not see how a proof in a stronger
system could be more convincing than a proof in the system itself.

If your worry
about PA is that it has "too much" induction in allowing induction for
arbitrarily complex formulae, then this proof could calm those worries
by restricting induction to quantifier free formulae (though allowing
"longer" inductions).

.

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