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

On Mar 21, 11:42 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 21, 2:53 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

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".

NO, NOTHING will follow, because it CAN'T HAPPEN. There is no action
of adding just a sentence to a language. The set of sentences of a
language is arrived at by INDUCTION (either first to get the set of
formulas, or, I guess, you could do the induction straightaway to get
the set of sentences). You can't just intrude arbitrarily into an
INDUCTION to throw in just a single sentence.

Why do you make such a
big deal of this?

Because you're using a completely LUDICROUS notion as part of your
argument.

I tried to make the point that what Godel did
amounts to precisely what you are ridiculing.

Then SHOW it. Please give the exact passage in his papers where you
think he does that.

Again: No first-order theory ever proves a tautlogoy. That is your
misunderstanding.

No, YOUR misunderstanding. Or at least it is your misunderstanding to
overlook that there are quite ordinary defintions of 'tautology' that
yield that every first order theory proves EVERY tautology in the
language of that theory.

There is no such thing as *a* tautology as far as
first-order languages are concerned.

WRONG. We have a perfectly rigorous definition of 'tautology' for
first order languages.

And such rigorous definitions only reflect the common notion that a
tautology is a sentence that is true under all truth table evaluations
of its sentential components. Under certain common definitions, what
is true or not is the SENTENCE and NOT the SCHEMA of sentences. All
INSTANCES of a schema have truth values, but the schema ITSELF does
not have a truth value. The schema itself is a formula that defines a
SET of sentences. It's not a SET of sentences that has a truth value,
but rather the sentences themselves.

Fine. I have no problem with this definition of tautology. The ;point
is that a theory like PA cannot say something about a SET of
sentences, because quantification over sentences is not permissible at
first order. So PA cannot prove that "Every sentence follows from a
contradiction". That is all I wish to empahasize here. Secondly if you
do permit quantificaiton over P and Q, then the sentence

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

does have truth value. Namely it is true, but such a truth must be
metamathematical with respect to a first order theory like PA, since
PA cannot ever say something about "all sentences". I hope you accept
this as a fundamental limitation of first-order logic. If you do,
please look at my reply to another post in which I raise the issue of
how Godel apparently managed to overcome this fundamental limitation
of first-order logic, using only finitary reasoning within first-order
logic (as claimed by Godelians; I do not agree with the "finitary"
bit). How on earth is this possible? Do you have any explanation?


First order languages can only
recognize instances of a schema representing a tautology.

The instances ARE tautologies. Moreover, though we may use schemata
for certain purposes, the definition of 'tautology' for first order
languages does not going through schemata.

The
tautology itself, say, P&~P->Q where P and Q are variables,

Not under certain ordinary definitions of 'tautology'. Under certain
ordinary definitions, if 'P' and 'Q' are meta-variables ranging over
formulas or over sentences, as the case may be, then  '(P&~P) ->Q' is
not ITSELF a tautology but rather it is a schema such that all its
INSTANCES are tautologies. However, of course, we're not always so
pedantic about that distinction so that we refer to '(P&~P) ->Q'  also
as a tautology even though, if we were precise, we would say it's a
schema such that all its INSTANCES are tautologies. (And I don't
preclude that there might be authors who define 'tautology' so that a
schema itself is a tautology, but rather I'm saying that given certain
common definitions of 'tautology', under those definitions, the schema
is not itself a tautology but rather its instances are.)


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

Regards, RS
.

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