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

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.

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

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