Re: Godel's theorem is invalid?


sradhakr wrote:
> This is the classical viewpoint, which has been disputed.

Never coherently.

> For example,
> I believe that Neil Tennant has asserted in his book ("The taming of
> the true"?) that all truths are knowable.

Then his definition of truth simply does not match the classical one.

> The issue here is whether one
> can meaningfully talk of "truths" that are *in principle* unknowable.

I don't know that anyone has ever alleged the existence of
such a truth. Truths that are in principle unprovable in PA or
ZFC are always going to be provable in some stronger system.
So if knowledge is being linked to provability then there simply
ARE NO truths that are "in principle unknowable". The first thing
you have to do to motivate THIS argument is find someone foolish
enough to allege that some particular mathematical thing is
"in principle unknowable". We can then thereAFTER dispute whether
there is any sense in which that "thing" could ALSO be a "truth".

> Please see my argument below.
> >
> > Imagine that you have some formula Phi(n). For example, n might
> > be the statement "2*n is a number that can be expressed as the
> > sum of two prime numbers". You can check to see that Phi(0) is
> > true, you can check to see that Phi(1) is true. For any n, you
> > can check whether Phi(n) is true.
>
> Let us take this real slow. What exactly do you mean by the assertion
> that
>
> "For any n, you can check whether Phi(n) is true"?

I mean that if you give me an n, I can check it, and it
doesn't matter what n you give me. And I won't need
infinite resources to check it, either. Some finite amount
of resources r(n) will ALWAYS suffice. r(.) might not
be recursive, though.


> What you obviously mean here is that you can check Phi(n) for
> infinitely many *instances* of n, taken one at a time

Right.

> and exhaust the class N of natural numbers.

Wrong. I am not claiming the ability to exhaust anything.
I am just saying that the "I can handle any 1" challenge
applies (exhaustively) throughout n. This does NOT mean
that *I* can exhaust anything.

> This is precisely what I am saying is meaningless.

Well, I'm sorry, you CAN'T SAY that.
The components have meanings. They combine
properly. I can in fact DO exactly this, so you can't claim
that the words that describe what I'm doing are meaningless;
I'm refuting claim *BY DOING* it.

> Every time you check Phi(n) for a given instance, there
> are infinitely many instances remaining to be checked; so you really
> cannot exhaust N this way.

Of course I can't, and I conceded that, and that has
NOT ONE IOTA OF IMPACT on the TRUTH or the MEANING
of what I originally said.

> So I don't accept that you can meaningfully
> make the above assertion,

Whether you accept it or not is irrelevant;
neither the meaning nor the satisfaction of the assertion
depends on your acceptance.

> unless and until what you say below is false:
>
> >Yet there may be no known way to check whether "forall n, Phi(n)" is true.

Classically there is simply NO inferential connection between
these two things. All the instances, for all n, even if we COULD
exhaust
them, could all come up true, and "forall n, Phi(n)" could STILL come
up
FALSE, because the "forall" INside the object language ranges over
MORE things than the "for all" out in the meta-language did.

> In other words, I dispute your assertion that "If one continues
> checking the truth of Phi(n), for each instance of n, taken one at a
> time, one will never ever find a counter-example" is meaningful;

Please; if it is false, that IS meaningful, it is just false,
and it is false IN VIRTUE OF its meaning, so you can't say
it doesn't have a meaning. The only possible way you could
be right in asserting that it is meaningless via THIS objection
is IF IT IS TRUE, but the fact that we are CONFIRMING ITS TRUTH
in this case means that it MUST have a meaning IN THIS CASE AS WELL.

So Shut Up.

> it is
> an infinite process that can never be completed *in principle*;

NO, it is NOT a process AT ALL, because we ARE NOT alleging
any TIME or any degree of difficulty in ANY of this. ALL the true
instances are true ALREADY, REGARDLESS of whether anybody
has "confirmed" or checked ANY of them YET. All the valid
proofs of all the instances exist ALREADY. No further process
IS REQUIRED. This boneheaded invocation of physicalist metaphors
is just that. We live in a place without time or place. DEAL WITH
THAT.

> what
> you are really saying is that we have some means of checking (proving)
> the truth of Phi(n) for an *arbitrary* (unspecified) n,

NO, I am NOT saying that. I am CONTRASTING this with that.

> which would
> obviously prove "For all n Phi(n)".

Obviously, but that's NOT WHAT WE'RE SAYING.

> Of course, classical logicians (you
> included) will not agree with me, but what I have said above actually
> holds in my proposed logic NAFL

Please. It CAN'T POSSIBLY hold.
It's not soomething you get to have an opinion about.
We are already talking about OUR logic.
Our logic actually has the property that infinitely many
instances of something can be individually provable.
If yours doesn't then good for you, but even there,
you can't deny that the concept is MEANINGFUL.

> and follows from its postulates.

I doubt that, frankly.

> > >Hence there *should* really be no distinction between (1) and (3),

Even if there shouldn't, there is.
If you know how to eliminate it, please
concentrate on coherently explaining YOUR
alternative logic instead of irrelevantly alleging
shortcomings in classical. Classical just is the way
it is; no value judgments required, and ESPECIALLY
no rewriting of the English dictionary ALLOWED.

> > > a
> > >distinction which classical first-order logic dubiously tries to
> > >maintain.

"dubiously" my ***.

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