Re: Godel cant tell us what makes a mathematical statement true

herbzet wrote:

Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
The aliases are here only to demonstrate to the readers that,
for *better* reasoning, we should base mathematical assertions
on syntactical provability rather than the old intuitive notion
of "truth". For example, to prove F is undecidable, with 100%
certainty, we should prove so using *only* syntactical rules
of inferences and axioms!
If a theory T proves a sentence is undecidable in T, then T is
inconsistent. (Not actually sure this is correct ("true") in
complete generality.)
Note that I said "we should prove" but didn't say "we can prove".
The point being is *in general* if you can't syntactically prove
F is undecidable, then no matter what else you might say using the
canonical definition of truth it wouldn't be sufficient. (And I can
demonstrate this, but perhaps in different post, if you'd like to).

But I've never said or believed we can always syntactically prove
an undecidability.

If one can prove in T' that there is a proof in T of the formula F,
then naturally the proof in T' should be completely formalizable.

Right. The proof in T' is just a normal proof, like countless other
1st order proof!

In fact, we wouldn't accept it _as_ proof if we didn't think
it was reducible to a completely formal proof. No argument
there.

What does "it" refer to, here? Nonetheless, I'd agree: a proof is a proof,
and no argument from me either!

That holds for any proof, come to think of it.

Sure: any proof is a proof!

What's the point of debate here though?

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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