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

On Mar 21, 1:45 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

GC&~GC&~TPC  

Note: you have apparently misread what I have written above.

Yes, I did. My "mind" saw a '->' where you wrote a '&'.

My apologies for my remarks regarding your sentential skills. It was
my lapse in reading.

But there's still no point to your argument. PA doesn't prove a
contradiction, so PA does not refute any instance of a schema of
tautologies.

I said IF PA proves a contradiction (see (**) above THEN PA would have
refuted the schema P&~P->Q

But PA does NOT prove a contradiction. So what is the point?

Sure, if  P&~P->Q is not in the language of PA, its formal negation is
also not in the language of PA. But all I am saying here is that PA
can prove itself inconsistent by proving a contradiction.

If a theory proves a contradiction, then the theory is inconsistent.

Sure it is. The schema P&~P->Q expresses that PA is consistent.

No it doesn't. The above is just a schema of tautologies. A tautology
does not express that a theory is consistent.

PA can
refute that schema by proving a contradiciton.

Yes, PA can refute an instance of that schema by proving a
contradiction. PA refutes EVERY sentence in the language of PA if PA
proves a contradiction. I said that already.

Note: I am not saying
that PA proves a contradiciton.

So what is your point in saying "If PA proves a contradiction
then ..."?

I am saying that PA can *in principle*
refute the schema P&~P->Q by proving the negation of one instance of
that schema.

If we define 'refute a schema' as 'refute an instance of the schema',
then sure.

But so what? PA does not refute any instance of the schema.

In order for that to happen, PA would have to prove a
contradiction and would therefore have to be inconsistent.

Yes, we KNOW that.

For that matter even ~(P&~P) or Pv~P, etc are
consistency statements for PA which are not expressible in the
language of PA and which cannot be proven in any first-order theory.

Okay, now you're just being silly for the sake of it, right?

(1) ALL sentences of the form ~(P&~P) or Pv~P that are in the language
of a first order theory are theorems of that first order theory (the
first in both classical and intuitionistic first order theories, and
the second in classical first order theories).

You are one confused guy. Just now we agreed that P&~P->Q or similar
schema are not within the language of first-order theories.

I said sentences "OF THE FORM". So you can retract your "one confused
guy" remark.

P&~P->Q is
a theorem scheme, i.e., there are infinitely many theorems of the form
P&~P->Q in first order languages, and every first-order theory must
prove all these theorems.

Yes of course. What is the point of all this?

But no first-order theory can prove 'P&~P->Q', where P and Q are *sentential variables*, >
because such a sentence
is not within the scope of a first-order language.

No, it depends on the precise treatment of the author.

If 'P' and 'Q' are variables of the meta-language ranging over
sentences of the object language, then, yes, no formula with them is
in the object language.

But if 'P' and 'Q' are sentence variables in the sense of variables in
the object langauge, then I think you may find that some authors allow
such a thing (I don't prefer it myself, and it's not common, but if
I'm not mistaken, you can run across it in the literature).

Anyway, *I* was the one to first mention here that if 'P' and 'Q' are
variables of the meta-language and not sentential letters (i.e., not 0-
place constants, we can specify) then no formula with them is in the
object language. So why are you harping back to remind ME of what I
made a point of myself?

I am saying that
sentences like these (which effectively involve quantification over
the P's and the Q's) are as good as consistency sentences for a first-
order theory like PA. Because PA would have to prove a contradiction
to refute such sentences.

Then I find that a quite odd notion of what a 'consistency sentence'
is. I don't see the point of it.

Under your notion of 'consistency sentence', in first order, any valid
(logically true) sentence is a consistency sentence.

What would be better to say is that "PA does not prove the negation of
a validity" is a consitency sentence. That gives us INFORMATION about
the theory PA. But we can simplify that to "There is a sentence in the
language of PA that PA does not prove" as a consistency sentence.

But just pointing to some valid sentence in the language gives NO
information as the consistency of any theory in that language. So a
valid sentence alone does not "express" anything about the consistency
of any theory.

You can assume I have done some "coding" or
"translation" to arrive at this conclusion.

Why don't you just define your sense of 'consistency sentence'? We see
that it subsumes 'valid sentence', but is it also co-extensional with
'valid sentence'?

MoeBlee
.

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