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



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.
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.

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

--
hz
.

Relevant Pages

  • Re: .EXE -> .ASM -> .EXE
    ... :this is a consequence of Godel's incompleteness theorem ... Consider the following classical examples of undecidability: ... which is inconsistent with being false. ... the truth or falsity of B is also "undecidable". ...
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  • Re: Godel cant tell us what makes a mathematical statement true
    ... canonical definition of truth it wouldn't be sufficient. ... an undecidability. ... it was reducible to a completely formal proof. ... You are arguing against the practice of giving intuitive, ...
    (sci.logic)
  • Re: proving that a statement is undecidable
    ... undecidability of his Goedel sentence, ... of axioms if those axioms cannot prove S and cannot prove. ... is a strictly stronger property than the property of undecidability. ... (truth in all models). ...
    (sci.math)
  • Re: Godel cant tell us what makes a mathematical statement true
    ... canonical definition of truth it wouldn't be sufficient. ... an undecidability. ... it was reducible to a completely formal proof. ... we must therefore examine the methods of the mathematician." ...
    (sci.logic)
  • Re: Godel cant tell us what makes a mathematical statement true
    ... on syntactical provability rather than the old intuitive notion ... canonical definition of truth it wouldn't be sufficient. ... an undecidability. ... we must therefore examine the methods of the mathematician." ...
    (sci.logic)