Re: Help writing a paper on Godel's Incompleteness Thorem

David C. Ullrich wrote:
> On 2 Dec 2005 14:59:50 -0800, "Scott" <ToaTerra@xxxxxxxxx> wrote:
>
> >Re Ullrich:
> >
> >Have another question for you. What would a proof have to look like -
> >to you - to refute Godel?
>
> You need to make up your mind. Are you refuting Godel or not?
>
> At first it sounded to people like you were claiming to have
> done so. Then you said no, we'd all agree that the proof is
> correct. If the proof is correct then there is simply no
> such thing as a refutation of it.
>
> Perhaps you don't actually mean to be talking about refuting
> Godel, but instead you're talking about building an alternate
> system of logic where Godel's theorem does not hold. If so
> then this is just one more example of a place where you
> don't say exactly what you mean - you _need_ to say _exactly_
> what you mean here if you want to be taken seriously.
>
> >I know you don't believe one exists, but what
> >are the areas/components you would expect to see present that you would
> >know it was a serious attempt?
>
> Again, no attempt at refuting Godel will ever be taken seriously
> by anyone who knows what they're talking about. If you're
> instead talking about building that alternate system, at
> minimum I'd expect to see the following:
>
> (i) Evidence that the writer has a clear understanding of
> the classical system and the major results in the classical
> system.
>
> (ii) Clear definitions of any new concepts.
>
> (iii) Actual valid reasoning regarding the new concepts,
> based on nothing but the definitions.
>
> (iv) Some evidence that the new system was interesting
> for some reason - in particular evidence that it was
> enough like traditional logic to allow us to do at
> least some of the things we do with traditional logic.
>
> All those are abundantly lacking in that pdf and the
> present exchange. To give just the first examples
> that spring to mind - there are many more:
>
> (i) You told us that we should beware that in that
> paper "true" and "provable" mean the same thing.
> If you think that they actually do mean the
> same thing you're showing incredible ignorance.
> If you don't actually think the two words mean
> the same thing then we're mystified why you use
> them interchangeably - you should write "true"
> where you mean true and write "provable" where
> you mean "provable".
>
> (ii) You give no definition of what "spatially
> separated" and "temporally separated" mean.
> I've asked what you mean by those terms and
> you have not replied. No, you're not going
> to be taken seriously as long as you use words
> without giving definitions of them.
>
> (iii) The "proofs" you give are not proofs at all.
> For example, you seem to think that if you read
> a certain proof that something is inconsistent
> and you show that that proof is invalid in
> a certain system then you have shown that that
> system is consistent. This is ludicrous - showing
> that a proof that A is false doesn't work does
> not show that A is true.
>
> (iv) Given that the system consists of just
> ignoring some things on Mondays and ignoring
> other things on Tuesdays (or so it seems to
> me, and you didn't say it wasn't so when I
> said that's how it seemed to me) it's really
> not interesting at all - avoiding inconsistency
> just by pretending it's not there is a little
> too easy. Also: If in fact Godel's theorem
> is not part of what's true on Monday then
> whatever it is that _is_ true on Monday
> must be missing _large_ parts of classical
> logic - you present no evidence that what
> _does_ exist on Monday is enough to be
> of any use whatever.
>
> ************************
>
> David C. Ullrich

A statement that's true in a theory is provable there, if the theory is
complete and consistent. If it's provable it is true if the theory's
consistent.

This Presburger arithmetic is said to be complete. Is that true, that
Presburger arithmetic is believed to be complete?

Ah, one notion that people studying fuondations of mathematics try to
understand is what would be the characteristic of a "true" axiom.
That's not necessarily about the most axiom-like or examplar of axioms,
it's about whether the statement of the axiom and all its application
is totally agreeable with all accepted facts, that it's true,
self-evident, and so on and so forth.

Then, the notion is to gather all the indepedent true axioms, where
independent means that they are not directly implied or theorems of
some of the other true axioms. Then, the minimal subset of all true
axioms would in some's opinion represent a theory to prove all facts
where it is complete. For an axiom set to represent a complete theory,
it must be so that every true fact about the objects defined in the
theory, every theorm, is a direct result of the axioms' implication.
Any fact is shown true, provably, in the complete theory.

Kurt Goedel, a logician, who is a person who studies logic with some
qualifications, in the early-to-mid twentieth century derived several
rresults that theories strong enough to be interesting, to prove
statements about general facts, are incomplete. There are several
related rresults. One is based upon the well-known Cantor's diagonal
argument. Another uses a function with 85 variables, which is a very
large number of variables. I note that most emphatically.

In modern times, Gregory Chaitin has taken the methodogy of Goedel even
further, going to thousand of variables in equation to calculate
approximately 64 bits of a constant that reflects whether or not a
random Turing machine wil halt, in a curious programming languge of
some sort necessarily, a UTM language, where Turing and Church are
contemporaries, in a simlar way as Goedel and Willard Quine are
basically contemporaries, and basically nemeses.

Goedel diligently set out to prove in his later years that there can be
no theory of everything. He was lucky beause that works well when
there's no universe in the theory, for example as is the case with the
standardized ZFC set theory, through no fault of its own or Zermelo and
Fraaenkel.

The first three paragraphs or so were really well-meaning and
expository but the last several became, not so much sarcastic, as
hyperbolic, true in interpretation, but just making fun of it.

That is not to reflect on Scott here's paper, or notions basically
having to do with the Liar paradox. Basically, Scott says there is a
way o encode each true statement, and only true statements, as
sentences or in some vague sense of computing programs. Then, he
notices each of those is basically an integer, except obviously there
are true sentences that are infinite. (Those vary in being
representable finitely or not, as most of them would be, i.e. in terms
of static program analysis.)

For example, last month or so M. Andrew Boucher, a regular contributor
to sci.logic, sent notice of a paper he had written to sci.logic.

http://groups.google.com/group/sci.logic/browse_thread/thread/6630260c08dffdee/

As a direct result of that dicussion, it was noted the Torkel is
absolutely humorless, except in the sense that that's a joke, and any
affiliation he carries is painted with the same brush. Anyways: there
is a set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ..., forever} and
there exists a set containing zero, and every element that is a
successor of an element of that set. On the one hand, it's an infinite
thing, on the other, as hugely complicated as shampoo instructions, or
BASIC's:10 X = X+1 20 GOTO 10.

So, Boucher posits a maximal element of sorts of the natural integers.
In doing so that's a universal element of sorts. In this model theory
under discussion, any arbitrary infinite ordinal is used. So anyways,
it was considered that the only complete theory is natural integers is
one with some maximal element, some universal element of the natural
integers as a compactification of comprehension, the sputnik of
quantification. Ahem. So, there is visible in these various threads
these convergent modes of reason towards universality in being and
logic.

Seventy years ago or so Goedel closed the book on the Hilbert program.
Maybe it's time to open that book again.

Ross

--
"The alternative, of a finite
number of scales and a lowest "layer of the onion" would,
when considered rightly, amount to no more than a giant
static diagram from Euclid, maybe with 2^10000 lines or
a number too large to print in the most compact notation
known to man if all the oceans were ink, but finite all
the same, and thus utterly preposterous!" - Ramsden

.

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