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

On Mar 22, 4:02 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
Obviously you need some kind of
coding to do this, and let us also accept Godel's translation
procedure as legitimate.Here is a step-by-step outline of the
difficulties that would follow:

1. Godel first costructed a metatheory T of PA in
whose language we
may construct and prove propositions about sentences of PA.

This is NOT relevant.
We are talking about modern treatments.
The relevant metatheory is model theory, IS ALWAYS set theory.
Or a class theory that is basically similar, like NBG.

2. The theory T is, in substance, precisely as strong as PA.

No, it isn't.
It's MUCH stronger because IT KNOWS THE DIFFERENCE
between finite and infinite sets. PA by contrast can have models
where some of the elements must in fact BE infinite, but PA can't
define that predicate.

I.e., we
may encode PA into the theory T, and nothing stronger than PA.

That is clearly impossible.
In PA, we can encode a lot of things, but your meta-
theory is UNdefined IF you are going to say "I'm using the
one Godel defined". Godel's meta-theory is not even important:
the result is about what you can say about PA *in PA itself*,
treating it AS ITS OWN meta-theory. Which particular meta-
theory you use to prove THAT is NOT important. ZFC will DO;
that is ALL *we* NEED to know!

3. Godel used his translation procedure to
translate the theorems of T
into the language of PA.

Please. T is already PA to begin with.
I really do not know where you are getting this
irrelevant characterization of what Godel did.

4. In particular, consider the following sentence in the language of
T:

(For all P) (For all Q) ((P&~P) -> Q) (*)

5. Let us say that (*) gets translated to the sentence S in the
language of PA.

If you can translate this accurately into PA,
THEN IT MIGHT AS WELL HAVE BEEN IN PA TO START WITH.
If P and Q are sentences of PA, then the mere existence of this
translation PROVES that this was in PA to start with, basically.


6. It turns out that S is provable in PA, as per my understanding.

7. However, the theory T is precisely as strong as PA,
which does not
prove the sentence (*).

This is INCREDIBLY stupid.
As you have defined things, (*) and S ARE THE SAME!
S is just the translation of (*) into the language of PA!
For you to say that PA proves S and does not prove (*)
IS RI*fucking*DICULOUS!
(*) is NOT EVEN IN THE LANGUAGE of PA!
SO OF COURSE PA can't prove it! But PA *does* prove S,
which as far as PA is concerned *is* (*).

So this whole thing is just dismissed as silly.


JEEzus.
I hope that clears THAT up.

(PA only proves all instances of (*), but not
(*) iitself).

It does, however, prove S itself. S is universally
quantified.

Therefore T shoudl not prove (*).

No, since S is the translation of (*) and PA proves S,
T *should* prove (*).


I really do not know why you are choosing to obsess over
"Godel's meta-theory T".
You only have 2 available choices for a meta-theory.
Either you can use PA itself or you can use ZFC.
What Godel used is NOT important.
He was still talking in terms of *PM*, for crying out loud.
Sometimes, Old IS *OLD*!
.

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