Re: My talk about Godel to the post-grads.

On Jun 28, 10:05 am, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
Aatu Koskensilta wrote:
"Nam D. Nguyen" <namducngu...@xxxxxxx> writes:

GIT is a *meta* theorem which is of course different than an ordinary
first order thereon, which in turn is syntactical in nature.

The incompleteness theorem is a perfectly ordinary mathematical
theorem. It is a "metatheorem" only in the sense that it has to do
with formal theories, but it is not in any philosophically significant
way different from any theorem in, say, finite combinatorics.

Depending on the exact phrasing, but one would find in some form
the phrase "true but not provable" in GIT! The word "true" will
differentiate it from a first order proof. And that's a *technical*
sense! Whether or not it's significant in whatever sense (philosophically
or technically) is a different matter.

"True" in that context can be precisely mathematically defined. Even
those versions of incompleteness that mention 'true' can be stated and
proven in a formal first order Z set theory, just as virtually all
ordinary mathematical theorems (or, for mathemtical theorems more
generally, also using a choice principle, which the proofs of the
incompleteness theorems do not use).

My post(s) basically alluded to that it can't be technically significant: it's
built on relativistic and incomplete (knowledge of) assumption of arithmetic.
In fact one could even say, technically speaking, FOL doesn't require
the concept of arithmetic: just that of "finite" (among other syntactical
concepts such as Rules of Inference, etc...).

So GIT does involve some first order theorems, but there's no way
it - itself - is an ordinary FOL theorem, let alone a "perfect" one.

It is provable in a perfectly rigorous first order Z set theory and in
even WEAKER perfectly rigorous first order theories. The
incompleteness theorem is provable on perfectly rigorous terms and
from assumptions even WEAKER than many ordinary mathematical theorems
as found in, say, analysis.

(Ask yourself where the word "true" would come from in all that
*syntactical* proofs? And I'm almost certain think you know that!)

'true' is perfectly, rigorously defined in the first order Z set
theory. Perfectly, rigorously we define 'true in a model', and
perfectly, rigorously we define a certain model of the language of PA,
which we call 'the standard model for the language of PA', and we then
we may use 'true' as elipsis for 'true in the standard model for the
language of PA'.

MoeBlee

.

Relevant Pages

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