Re: Godel's comments about the "true reason" for incompleteness

On Mar 21, 6:18 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
On Mar 22, 4:12 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:> On Mar 21, 3:46 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

I do not have any essential disagreement over what you have said
above. Thanks for the clarification.

Okay, good. So we got tautologies worked out.

Sure, The main point to which we both agree on is that a first-order
language cannot express sentences like "All sentences follow from a
contradiction".

No, I NEVER said ANYTHING like that!

And the main point was NOT that. The main point, which I belabored
over and over, is that (ordinarily) a schema of tautologies is not a
tautology but rather each INSTANCE of the schema is a tautology and
that first order theories certainly DO prove every tautology in the
language of the theory.

The infinitely many sentences in this schema may all
be tautologies proven by the object theory, but the object theory
cannot formalize the notion of "all sentences", at least not without
the coding employed by Godel.

It DEPENDS on the object theory T and whether "all sentences" refers
to sentences of the T's OWN language or of some object language T*
that is also a meta-theory.

For example, Z set theory is a first order theory. And it's an object
theory. AND it's a meta-theory for certain object theories. And in Z
set theory as meta-theory for PA we may discuss all kinds of things
about sentences of PA.

So the truth of the assertion that "All sentences follow from a
contradiciton" would have to be a metamathematical truth *about* the
object theory. At least prior to Godel this was the situation. The
claim that such a meta-sentence can be translated into a sentence in
the object language is what iI am contesting.

Define what YOU mean by 'translated'. And you can "contest" all you
want; meanwhile, though, you've shown no contradiction or even
intuitive problem with incompleteness proofs.

Such a translation
cannot be via purely finitary first-order reasoning.

Please define what YOU mean by 'translation'.

MoeBlee
.

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