Re: Update: Objections to Cantor's Theory



david petry wrote:
> I'm starting a new thread because the other one got
> out of control.

All threads about this are inherently out of control.
There is no there there.


> I want to thank those of you who gave me some useful
> feedback on my article (you know who you are :-).

Of course we do, but it's not clear that YOU do, since
you persist. I in particular replied late and it is not
clear that you were still paying attention.


> Here's the updated version of my article. I hope it
> answers some of the questions people have asked.

It doesn't answer the only question *I* asked, so I will
have to repeat myself.


> ********
> In this article, "Cantor's Theory" refers to the pre-formal
> ideas about set theory introduced by Cantor in the latter
> part of the nineteenth century.

NO, IT DOESN'T.
Cantror's Theorem is provable in perfectly standard formal
first-order set theories like ZFC and NBG. It is proof of
how ignorant the objectors are that they can't even recognize
THESE proofs AS proofs.

> The "anti-Cantorians" are
> people who claim that Cantor created a fantasy world.

Absolutely ALL of first-order logic (to the extent that
it is infinitary and symbolic) is a fantasy world!
Have you ever seen or observed the letter "a", or the digit
"0"? OF COURSE NOT! You have seen particular individual
physical tokens of characters; you have seen MANY DIFFERENT
a's and many different 0's, but you have NEVER ONCE seen
THE symbol 0 or THE symbol "a", or even THE word "David"!
THESE THINGS ARE ALL *ABSTRACT*! Saying about anything that
it is "a fantasy world" is just MEANINGLESS in this context.
OF COURSE it's fantasy, to the extent that it is not concrete.
Neither Zero nor One nor COUNTABLE infinity is ANY more concrete
than these higher infinities that the idiots are objecting to!

> The anti-Cantorians claim that while infinite sets and
> power sets of those infinite sets are undeniably useful
> abstractions,

They claim nothing of the kind. If this were the case then
the argument would simply be over and they would have simply
lost.

> what Cantor did was to take an argument
> (the diagonalization argument), which is perfectly valid
> in concrete mathematics,

It is a LOGICAL argument and it is valid in the context
of FIRST-ORDER LOGIC. This has absolutely NOTHING to do with
any sort of distinction between "concrete" and "abstract"
mathematics, a distinction which YOUR ignorant ass is FAR
from competent to define IN ANY case! ALL math is abstract
BY DEFINITION!

> and wrecklessly apply it to
> the abstractions of infinity,
> ultimately producing garbage.

This is ridiculous. There are an infinite number of natural numbers.
If you alleged that every last one of them had a double, or a square,
would you be doing something "wreckless"? That's spelled "reckless",
by the way. If you are doing a finitary simple operation (such as
taking
half of a rational number) and you can do it to ANY member of some
class,
then the fact that the class is infinite SIMPLY DOES NOT MATTER. There
ARE times when it matters but THIS IS NOT one of them.

Diagonalization IS A TURING-MACHINE argument, in ADDITION to a
Cantorian
one. TMs are infinitary BY DEFINITION: They MUST have infinitely long
tapes and they MUST admit NO upper bound on the length of their input
beyond that it be finite. So EVERY TM has AN INFINITE number of
possible
inputs and can represent a potentially infinite class of the
corresponding
outputs, if it halts on enough of them. Given that TM programs can
themselves
be encoded as inputs to TMs, ANTI-DIAGONALIZATION IS UNAVOIDABLE. YOU
CAN WRITE A SHORT TM program to anti-diagonalize ANY AND EVERY TM.
The fact that the idiots think we are applying this TM "wrecklessly"
does
NOT in ANY way compromise its VALIDITY as a TM. And no TM produces
"garbage",
no matter HOW it is applied: it either halts or it doesn't: THAT IS
ALL.


> The Cantorians (i.e. almost all pure mathematicians) claim
> that since Cantor's Theory can be formalized in a logically
> consistent way (e.g. ZFC), and since the study of
> formalizations is certainly a part of pure mathematics, there
> is absolutely no room for debate about Cantor's Theory.

This is backwards. ZFC does not formalize "Cantor's Theory".
ZFC, precisely because IT IS a formal first-order theory,
is bigger, better, and badder than Cantor's Informal Theory
COULD EVER HAVE HOPED to be. Moreover, this simple fact alone
entitles ALL of ZFC's theorems to the utmost respect UNTIL AFTER
somebody PROVES A CONTRADICTION from ZFC! UNTIL THEN, ALL the idiots
are PERFECTLY welcome to SIT DOWN AND SHUT UP, not only about Cantor's
theorem, but about EVERY theorem of ZFC that is not itself a deep
axiom (like Choice).

> When Cantor introduced his ideas, there was a heated debate
> about whether they should be accepted as mathematics. For
> reasons which are not entirely clear, the ideas were
> accepted,

Please. They were clear to Zermelo. They were clear to Hilbert.
They were accepted because nobody has ever derived any contradiction
from the axioms positing them. In math, that is ALWAYS *enough*!
The fact that it is not enough for the idiots just proves they're
idiots.

> and the debate has fallen silent within the
> mathematical literature. However, the debate has flaired
> up again on the internet.
>
> Most of the debate on the internet about Cantor's Theory
> is junk. The topic is a crank magnet. Most of the people
> who participate in the debate, have no deep understanding of
> the issues. However, hidden within all the noise, there does
> seem to be some signal.

No, there doesn't, and you're flaunting your own ferrosity
by suggesting otherwise.

> While the pure mathematicians almost unanimously accept
> Cantor's Theory (with the exception of a small group of
> constructivists), there are lots of intelligent people who
> believe it to be an absurdity.

No, there are not.

> Typically, these people
> are non-experts in pure mathematics, but they are people
> who have who have found mathematics to be of great practical
> value in science and technology,

Possibly.

> and who like to view
> mathematics itself as a science.

No, I'm sorry, they do nothing of the kind.
They do not know enough philosophy to even know
HOW to do THAT. They view math as something completely
other than what it actually is, possibly, as you said before,
as computation. But computability is every bit as diagonalizable
as powersetting, so that doesn't help.

> These "anti-Cantorians" see an underlying reality to
> mathematics, namely, computation. They tend to accept the
> idea that the computer can be thought of as a microscope
> into the world of computation, and mathematics is the
> science which studies the phenomena observed through that
> microscope. They claim that that paradigm

WHAT paradigm??
Last I heard, around here, we, being logicians, were
tempted to view the computational paradigm in terms of
Turing Machines. In that case, the idiots get no help,
because first-order reasoning CAN be formalized via the
TM paradigm. Moreover, the TM paradigm itself supports
(anti-)diagonalization! So saying "it's all about computation"
affords the idiots NO HELP WHATSOEVER.

> encompasses all
> of the mathematics which has the potential to be applied to
> the task of understanding phenomena in the real world (e.g.
> in science and engineering).

OK, fine. That still doesn't help them.
That paradigm shows that if every"thing" is computable, then
some"thing" different from all those"things", is, by diagonalization,
ALSO COMPUTABLE. THIS IS A CONTRADICTION. Therefore there
exist NON-computable "things". There is NOTHING they can do about
this.

> Cantor's Theory, if taken seriously, would lead us to believe
> that while the collection of all objects in the world of
> computation is a countable set, and while the collection of all
> identifiable abstractions derived from the world of computation
> is a countable set, there nevertheless "exist" uncountable sets,

Replace "uncountable" with "uncomputable" and you can just prove this,
both *about* TMs AND *with* TMs.

> implying (again, according to Cantor's logic)

Please. The only LOGIC relevant here is STANDARD
CLASSICAL FIRST-ORDER LOGIC! There is NOTHING
Cantorian about THAT! This is just logic, PERIOD!
THAT is what the idiots don't understand!
That's WHY we're justified in dismissing them as idiots!

> the "existence"
> of a super-infinite fantasy world having no connection to the
> underlying reality of mathematics.

There IS NO "underlying" reality of mathematics, unless you mean
precisely this fantasy world. Mathematics is ALL science fiction.
It's about WHAT IF *this* were axiomatically true?

> The anti-Cantorians see
> such a belief as an absurdity (in the sense of being
> disconnected from reality, rather than merely counter-intuitive).

Again, ALL axioms are like that.
0<1
is just 3 symbols. You cannot point to a physical 0 or 1
that they could be about. "<" does NOT have to connote being
to the left of something on a number line. It does not have
to be ANYthing concrete. ALL of this is abstract.
ALL of it is disconnected from "reality". JEEZUS.


> The mathematicians claim that they can "prove" the existence
> of uncountable sets, and hence there's nothing to be debated.
> But that merely calls into question the nature of "proof".

No, actually, it doesn't. We can say very unambiguously and
finitarily EXACTLY what we mean by "proof" (proof-predicates are
PRIMITIVE-recursive). At best it calls into question the
EXISTENTIAL IMPORT of proof. This is where you need to say something
about the Lowenheim-Skolem theorem. All these theories that prove
the existence of uncountable sets can (at first-order) be modeled
in a countable universe. This really does mean in some sense that
these proofs DON'T (by themselves) necessitate the existence of
uncountable sets. They do, however, given any alleged counting,
necessitate the existence of something that it leaves out. The
problem is that that's only ONE (as opposed to uncountably infinitely
many)thing left out.

> Certainly infinite sets and power sets exist as absractions.

I repeat, in that case, the cranks SIMPLY LOSE. EVERYthing being
talked about here is "abstract". YOUR NAME is abstract.


> But, abstractions don't necessarily obey exactly that same
> laws of logic as directly observable objects.

The level of ignorance you are flaunting here is breathtaking.
First of all, "directly observable objects" don't obey ANY LAWS
OF LOGIC, PERIOD. Logic IS abstract and it is ABOUT abstractions.
ONLY abstractions obey ANY laws of logic at all!

> Assuming
> otherwise can turn abstractions into fantasies, and proofs
> into absurdities, and that's the crux of the anti-Cantorian's
> argument.

Then they simply have no argument. There is no difference
between an "abstraction" and a "fantasy" TO BEGIN WITH, so
there is nothing to turn. Moreover, EVERYthing "concrete"
or non-abstract is UTTERLY AND COMPLETELY IRRELEVANT to this
WHOLE ENTERPRISE ANYWAY! The fact that the cranks surmise otherwise
is one thing that makes them cranks.

> The pure mathematicians tend to view mathematics as an art
> form. They seek to create beautiful theories, which may happen
> to be connected to reality, but only by accident. Those who apply
> mathematics, tend to view mathematics as a science which explores
> an objective reality (the world of computation).

I already rebutted all this in my previous reply in the other
thread, which you probably haven't read. You need to.

> In science, truth
> must have observable implications, and such a "reality check"
> would reveal Cantor's Theory to be a pseudoscience;

Quite the contrary: it reveals that it has a model, so the
cranks may sit down and shut up, at least until after
they have understood the Lowenheim-Skolem theorem.

> many of the
> formal theorems in Cantor's Theory have no observable implications.

This is simply a lie.
All theorems following from an axiom set have the OBSERVABLE
implication
that NO model of the axioms can decide the theorem falsely.

> The artists see the requirement that mathematical statements must
> have observable implications as a restriction on their intellectual
> freedom.

Here, you are just flaunting your personal ignorance.
"Observable implications" is not just oxymoronic, it's moronic.
In my previous reply, I gave you the difference between an observation
and an implication. I repeat, go look it up.

> The "anti-Cantorian" view has been around ever since Cantor
> introduced his ideas. Witness the following quotes from
> contemporaries of Cantor:

This is the most intellectually dishonest thing I have ever
seen here. Those contemporaries were certainly in a completely
different mindset from any modern thinkers, and Kronecker in
particular utterly lacks credibility in this whole context.


> "I don't know what predominates in Cantor's
> theory - philosophy or theology, but I am sure
> that there is no mathematics there" (Kronecker)

THAT is NOT a derivation of a contradiction from any of
Cantor's axioms.

> "Set theory is a disease from which mathematics
> will one day recover" (Poincare)

And THAT is not specific to Cantor.


> In the contemporary mainstream mathematical literature, there
> is almost no debate over the validity of Cantor's Theory.

Nor is there even any acknowledgment OF THE EXISTENCE of such
a thing. INSTEAD OF "Cantor's Theory", WE NOW have all the
THEOREMS following from AXIOM-SETS like ZFC and NBG. It is THESE
that don't need debating (because they HAVE been PROVEN)!
Cantor's TheorEM, NOT "Cantor's Theory", is what is relevant NOW.

> However, some mathematicians still drop hints that they see
> the absurdity of Cantor's Theory. Consider:

> "Set theory is based on polite lies, things we agree
> on even though we know they're not true. In some ways,
> the foundations of mathematics has an air of unreality."
> (William P. Thurston)

That's a much better quote. But it's
ridiculously hubristic. William P. Thurston in point of
actual fact doesn't know ***, let alone that the axiom of
choice or the continuum hypothesis "is not true".
At best he can say there are some models in which they are not
true. But there are others in which they are. He's got no
more clue than anyone else which side to pick, or should I say,
which side "the truth" has picked.



> It was the advent of the internet which revealed just how
> prevalent the anti-Cantorian view still is;

It has revealed no such thing. Have you never heard of "spam"?
A VERY SMALL number of people can create a VERY LARGE percentage
of the traffic. The percentage of the messages here that are about
this topic IS NOT in any relevant proportion to the fraction of
thinkers in the field who take that viewpoint seriously.

> there seems to be a
> never-ending heated debate in the Usenet newsgroups sci.math and
> sci.logic over the validity of Cantor's Theory.

No, there doesn't. There seem to be perennially recurring individuals
who bring it up. They come and they go. Herc is momentarily gone,
thank God.

> Typically, the
> anti-Cantorians accuse the pure mathematicians of living in a
> dream world, and the mathematicians respond by accusing the
> anti-Cantorians of being imbeciles, idiots and crackpots.
>
>
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation),

Well, I hate to break this to you, but (anti-)diagonalization
IS EVERY BIT AS OBSERVABLE "in the world of computation" as it is
in the proof of Cantor's Theorem.

> and philosophical mathematics, wherein Cantor's Theory is merely
> one of many possible formal "theories" of the infinite.

Even that is over-claiming; as far as math knows, there simply IS NO
"THE" Infinite for ANYthing to be a formal theory OF. Every theory
is different. Each one is going to be about a DIFFERENT infinity.

.

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