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

On Mar 21, 3:45 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
Re: Mar 20, 10:55 am, "R. Srinivasan" <sradh...@xxxxxxxxxx>:

After having responded to that post, I think it would help to focus on
just, say, two of the most obviously ridiculous things in it:

Now think of the following situation. Let us add P&~P-->Q (and only
this single sentence, where P and Q are arbitrary) into the language
of PA,

Please get back after you've consulted a logic book to see that it
makes no sense to say that we add just a single sentence to a
language.

Of course I said "inconsistency will follow". Why do you make such a
big deal of this? I tried to make the point that what Godel did
amounts to precisely what you are ridiculing. He brought in a sentence
like P&~P->Q (note: I am talking about the sentence with P and Q as
variables) into a first-order language via translation, call this
sentence S say, in the language of PA. Note again: Godel translated
the *schema* P&~P->Q into a single sentence in the language of PA. But
the logic is still the same, the proof principles are still the same.
So such a sentence should be neither provable nor refutable in any
consistent first-order theory (using only finitary principles),
including PA. *I* am asserting that as self-evident. No first order
theory that uses only finitary principles can ever establish the
truth of a schema via a single proof, either directly or indirectly
(e.g via Godel's translation procedure).

If you want to claim that PA proves S and S is the translation of the
*schema* P&~P->Q, then you have *obviously* stepped beyond the
boundaries of first-order logic at the translation stage. I am
rejecting the translation procedure employed by Godel as infinitary
(there are other reasons for this as well).

By the way, what precisely is the metatheory in which this translation
is carried out? My understanding is that Godelians claim that this
metatheory also uses nothing stronger than finitary reasoning which
can be carried out in a weak theory of arithmetic. That is as good as
claiming that P&~P->Q is a legitimate sentence in the language of such
a theory (say, PA), as *proven* by Godel's translation procedure
carried out using nothing stronger than PA. I do not accept this
because in my view such a translation must use infinitary priciples
that *obviously* step beyond the boundaries of first-order logic,
since such ;principles have managed to legitimize a sentence that is
out of bounds for a first-order language.


If
P&~P->Q has been translated to a formula S in the first-order language
of PA, and if PA proves S, are you denying that PA has *effectively*
proven itself consistent?

Please get back after you've consulted a book on logic to see that a
theory proving a tautology (or even just a contingency) does not
entail that the theory is consistent, and not anywhere in the
imagination of anyone but the most bitterly confused does it entail
that the theory has ""effectively" proven ITSELF consistent".


Again: No first-order theory ever proves a tautlogoy. That is your
misunderstanding. There is no such thing as *a* tautology as far as
first-order languages are concerned. First order languages can only
recognize instances of a schema representing a tautology. The
tautology itself, say, P&~P->Q where P and Q are variables, is NOT
provable in any first order theory, at least not until Godel came
along and managed to translate a *schema* P&~P->Q into a single
sentence S in the frst-order language of arithmetic (say, PA).

I am asserting that the sentence P&~P->Q (note that this is not a
first-order sentence, it has variables in it) or say, ~(P&~P), etc.,
which are tautologies, are actually metamathematical truths with
respect to first order theoreis (say, PA) that prove every instance of
the corresponding schemas, *assuming* that such theories are
consistent. The only way for a first-order theory (say, PA) to refute
a tautology (note: the sentence with variables in it) is by proving a
contradiction, and that is possible only if such a theory is
inconsistent. That is why I assert that a sentence like P&~P->Q, with
variables in it, is a consistency sentence and if Godel managed to
translate this sentence into a single sentence S in the language of a
first order theory (say, PA) then S is a consistency sentence for PA
as far as I am concerned.

You don't even begin to understand what I am talking about here. Try
again.

Regards, RS
.

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