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

On Mar 21, 3:24 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

You are right when you say that "There is no
tautology in the language of a theory that is not a theorem of the
theory". Because all tautologies are theorem schemes, not theorems,

WRONG WRONG WRONG. A tautology is a formula that is tautologically
valid. And every theory has as theorems all sentences of the language
that are tautologies.

and are metamathematically true,

They're LOGICALLY true, thus, a fortiori, mathematically true.

As far as I am concerned, anything that is asserted as true but which
is not provable in first order theories is metamathematically true.

But all logically true formulas in the language of a first order
theory theory ARE provable in the theory.

The point I am making is very simple. Consider the theory PA and let
us take (P&~P)->Q, for a fixed P. The assertion that every sentence in
the language of PA (as represented by the variable Q) follows from the
contradiction P&~P is not provable in PA.

Hmm, is that correct even if coded? But, okay, I'll play along...

While all sentences
represented by this schema are tautologies, PA does not recognize the
notion of "all sentences" and so cannot prove that "All sentences
follow from a contradiction". So as far as I am concerned, this is a
metamathematical (or metatheoretical if you like) truth *about* PA,
but not something that is provable *in* PA. Do you agree?

I certainly agree that in uncoded form, the assertion "Any sentence in
the language of PA can be proven from any contradiction stated in the
language of PA" is in a meta-language for PA and not in PA. So what?
That doesn't entail that there aren't sentences in the language of PA
that are true iff [fill in some meta-statement here]. Though, the more
famous is case is the other way around in this sense: The famous
sentence G is in the language of PA, and G is true iff G is not
provable in PA (the meta-sentence there is "G is not provable in PA"),
and G is not provable in PA.

The point I have been making all along has gotten lost in the
wrangling. It is the following. Post Godel, we find that PA does,
after all, *prove* that "All sentences follow from a contradiction
P&~P", contrary to what I have asserted in the previous paragraph. See
the following thread in which Aatu pointed this out to me and my
objection to the same:

http://groups.google.la/group/sci.logic/msg/cc1af3d1f93ca3f1

I would take it you are referrring to the fact that PA proves certain
assertions that is true iff [fill in a certain meta-sentence here] (I
hope I'm stating that correctly). That was NOT lost in the wrangling.
I mentioned it TWICE in one of my previous posts, and you just skipped
responding. So, for your particular example, perhaps what you're
finiding is that there is a sentence, call it 'E', in the language of
PA such that (1) PA |- E, (2) E is true, (3) E is true iff the
explosive rule holds for PA. (or something like that, since at this
point I'm really not interested in poring over your old posts).

But so what? PA does not prove sentences not in the language of PA.
But PA does prove certain sentences in its language that are true iff
[fill in a certain meta-sentence here]. That is not a contradiction.

I am asserting that no first-order theory using only finitary
principles, like PRA or PA, can ever prove something about "all
sentences". This is a fundamental limitation at first order and we set
up first order logic accepting this as a limitation. Having done that,
we cannot contradict ourselves and accept that PA does after all,
prove that all sentences follow from a contradiction.

Please just post the SPECIFIC sentence in the language of PA that you
are referring to and then the specific contradiction, in whatever
language (PA or meta-language), you think is a theorem, of whatever
theory (PA or the meta-theory).

Please do this so that we can move beyond your customary vagueness and
address whatever concern you have exactly.

Godel apparently managed to translate the entire set of tautologies
represented by the schema P&~P->Q into a single sentence S in the
language of PA.

You still have not said what you mean by "translate" let alone
"translate a SET". It is irritating talking to someone such as you who
so tenaciously persists to be as vague as possible at every juncture.

Obviously S must quantify over sentences (indirectly,
via coding) and is provable in PA. That is the only way in which PA
can prove that "All sentences follow from a contradiction". I am
asserting that PA should not prove S because we already agreed that PA
cannot prove that "All sentences follow from a contradiction" while
formlating first order logic..

AGAIN, PA can't prove what is not in its language. But that doesn't
obviate that PA can prove things in its language that are true iff
some other sentence in the meta-theory holds.

You see such an affair as a contradiction the way my neighbors who
were in the John Birch Society saw communists hiding in every dark
corner of their own home.

Anyway, PLEASE (and AGAIN, PLEASE) post your SPECIFICS before coming
back to post more of your vague confusion.

MoeBlee
.

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