Re: Godel Contradiction

On May 30, 10:46 am, slartibartfast <tomokane2...@xxxxxxxx> wrote:
IF T* doesn't include G(T) as an axiom then T* would have to include
other axioms that would allow the derivation, in T*, of E [ G(T) ],.
And so the proof G(T) would rest on the acceptance of these other
axioms instead.

Correct.

Now, here's the juicy part: Our T* can be proven in a theory as
limited as primitive recursive arithmetic (PRA) and using only
intutionistic logic. (I don't know the details of proving that - it
involves some fairly sophisticated "coding" - but you can investigate
that matter in literature.)

if you don't know the details of proving that then on what basis do
you assert it as true.?

I don't, to a mathematical certainty as verified by my own witness of
the proof steps, assert that it is true. On the other hand, given that
I know of no challenge in the literature to the quite common statement
that such a proof does exist, I think it is reasonable to surmise that
it does. I don't insist that you surmise that also; it's entirely up
to you. Meanwhile, as to this particular matter I represent only that
I am reporting an ordinary (and, as far as I know, unchallenged) claim
in the literature. Regarding certain other claims, I have verified
them by my own inspection of each proof step; but my knowledge in that
way, of course, does not include ALL of the published and ordinarily
accepted theorems in the literature.

In other words, we can prove the
incompleteness theorem in a theory T* that pretty much assumes
nothing
more than algorithmic arithmetic done constructively.

now HERE'S where one of us is getting confused:
BECAUSE...
IF T* is PRA plus a few other axioms, then by your logic, T* can prove
E [ G(T*) ] since a subset (PRA) of T* can.

My understanding (and I welcome knowing any particular important
technical qualifications that I may be omitting) is that PRA using
only intuitionistic logic (T* in this case) proves that if T (say, PA,
in this case) is consistent then PA |/- G(T).

And I didn't say anything about "plus a few other axioms".)

Also, I don't know what your notation "E[G(T*)]" is intended by you to
mean. I suppose it means "there exists a sentence G(T*)". But that is
not what is at stake. Rather, the proof (in arithmetic) merely
mentions a certain (Godel) number and shows that number to have a
certain arithmetical property. Then we show (in a meta-theory) that
that number has that arithmetical property iff the consistency of the
object theory entails that the sentence CODED by that number is not
provable in the object theory (as well as we prove in our meta-theory
that certain object theories, such as PA, are indeed consistent, so
that the aforementioned sentence is not provable in the object
theory).

Also, you refer to a subset (PRA) of T*. That is correct, in the
trivial sense that any set is a subset of itself.

And it seems you're driving at the result that a theory proves
something about itself. I'll address that in remarks below as well as
my remarks above address this matter.

and that just sounds like self-contradictory gibberish to me.

But it's not. I can't control that something that is correct sounds
like "self-contradictory gibberish" to you in particular. And your
claim about that is ironic, given that I am the one cleaning up your
own, at best, idiosyncratic formulations.

i.e what if PRA is a subset of T* ?

Since PRA(I) (PRA with only intutionistic logic) IS T*, trivially, PRA
(I) is a subset of T*.

Moreover, we may ask you, "From what axioms do you ordinarily accept
as proving any of various mathematical theorems, ranging from "1+1=1"

interesting new theorem there MoeBlee !

Thank you, it took me 23 years - virtually a lifetime - to prove it! I
do think it will revolutionize mathematics. Check out Hilary Putnam's
article "MoeBlee's Idempotency Of The Unit Thesis In Confabulated
Pseudo-Arithmetical Counter-Structures" in the 'Journal of the Annals
of the Bulletin of the Proceedings of the 3rd Annual Conference of
Advanced Inner Mathematics'.

So why in the world would you accept a
whole bunch of mathematical theorems as proven but not the
incompleteness theorem that can be proven from an even WEAKER set
of
assumptions than those used to prove all the mathematical theorems you
accept?

That the incompleteness theorem CAN "be proven from an even WEAKER > set
of
assumptions than those used to prove all the mathematical theorems I
accept" I only have so far by your ipse dixit

If you don't believe my report, then why not just look it up in many a
discussion of the matter in the literature?

and the reason I don;t
yet accept it is because of a seeming contradiction in T* proving the
incompleteness theorem for itself using that subset of itself called
PRA.

So your objection is to the result that certain theories prove the
incompleteness result about themselves.

(1) Recall that the incompleteness result is NOT "theory T is
incomplete" but rather "IF theory T is consistent then theory T is
incomplete". (2) The process involves Godel numbering, and the theorem
itself might not mention "theory T" but rather the theorem IN theory T
is an arithmetical sentence that we show (in our meta-theory) to be
true (recall 'true' means 'true in the standard model of PA' in this
context) iff the consistency of theory T entails that theory T is
incomplete.

(I welcome any suggestions if my synopses on these matters is not
correct or requires important qualifications.)

So what EXACT contradiction do you claim to derive? The claim that a
particular T (or any theorem T of a certain kind) proves a certain
theorem can be formalized and proven (or not proven, as the case may
be) in such a theory as say, Z set theory. So if there is a
contradiction to be derived about what we prove, then that
contradiction is of the form of sentences P and ~P that are theorems
of Z set theory. If you claim a contradiction, then you need to to
present such a sentence P and your argument as to why both it and its
contradiction are theorems of Z set theory.

/

Finally, what system, axioms, or principles of mathematics do you
accept as basis for proof of the various mathematical theorems that
you accept?

Even IF such a limited system as PRA(I) does not prove the
incompleteness result, the result is provable easily from, say, Z set
theory, in which also ordinary mathematics can be formulated and
proven. But suppose you don't accept Z set theory; fair enough, but
then what system, axioms, or principles DO you accept as basis of
proof? Then we can ask whether that system, axioms, or principles also
prove the incompleteness result. (But, anyway, I don't think you'll
find it controversial that the incompleteness result is proven from as
limited a system as PRA(I), as, meanwhile, it would be curious that
you'd reject PRA(I) which would seem to be about as conservative as a
basic mathematics can be.

MoeBlee

.

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