Re: Cantor's diagonal proof wrong?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/17/04 

Date: 17 Nov 2004 11:28:10 -0800

greeneg@cs.unc.edu (George Greene) wrote in message news:<992b156f.0411161714.6e4b426f@posting.google.com>...
> raf@tiki-lounge.com (Ross A. Finlayson) wrote in message news:<3c6b9c1e.0411161115.d4c1d84@posting.google.com>...
> > In a _very simple_ set theory, each ordinal is a set, and each set is
> > an ordinal.
>
> That is arguably not simple enough.
> The usual set-theoretical successor of x is xU{x}, NOT p(x).
> There are GOOD reasons for this.
>
> > The powerset is just the order type, which is just the
> > successor. The function mapping ordinal to successor is f(x)=x+1,
>
> That is definitional if you are going to have ordinal arithmetic at all.
>
> > Hey, people besides Curt, why is it that the only thing that anybody
> > on sci.math seriously argues _against_ is the diagonal theorem?
>
> That is an oversimplification. There is a FAQ. LOTS of things
> get argued against. I would say that one reason why this one is
> prominent is that we get a lot of arguers who don't know basic
> first-order logic and who don't know what the relevant axioms are
> if they're going to be talking about powersets. They have
> seen an "intuitive" proof and want to keep reasoning in that
> vein. What Cantor's Theorem REALLY says is "you can't biject a
> set with its powerset". Given any set theory, that proof is very
> straightforward and neither you nor anyone else would know how
> to attack it -- and it bears stressing that it applies not only
> to every set, but every CLASS as well, NO MATTER HOW BIG.
> AND, for that matter, no matter how small: Finite/infinite
> ISN'T EVEN RELEVANT -- it holds for all finite
> sets too, no matter how small, including the empty set.
> The problem is that people have only been exposed to the
> one particular case, and they get distracted by the details
> of that case. The REAL truth is simply that if you understand
> why Russell's paradox is paradoxical, then you understand the
> proof of Cantor's theorem. If you were cursed to have been originally
> exposed to some proof that hides the connection, well, don't blame us.
>
> > Where there's smoke, there's often fire.
>
> BUt not here. Here, there is just a lot of irrelevant distraction
> because people think w or aleph_0 is special. With respect to
> Cantor's theorem, IT ISN'T. It is JUST LIKE EVERY OTHER set in
> being non-bijectible with its own powerset.
>
> > Several on this thread are quick
> > to defend "mathematics that has no relation to any known aspect of
> > physical reality." Transfinite cardinals are mental masturbation, a
> > house of cards, a valiant effort to make reason out of inconsistency
> > that is doomed.
>
> HTF would you know?? Don't you see that the burden of proof IS ON YOU?
> IF ZFC + large_cardinal_axiom_of_RAF's_choice REALLY IS
> inconsistent, THEN THERE IS A FINITE *PROOF* of that.
> If you are telling the truth then you have NO EXCUSE for not
> PRODUCING this proof! No, we are not holding our breath.
>
> > I laugh about the pickled three-headed sheep comment.
> > Counterintuitive is one thing, reasoning counter to the obvious
> > infinite nature of some infinite sets is flawed reasoning.
>
> No, it isn't. Believing in "obvious nature" that you can't
> axiomatize, and THEN calling THAT "reasoning", is brute stupidity.
>
>
> > Paradoxes are wrong! Pairs of dachshunds are great, although they're
> > yippy little dogs, tastes vary, paradoxes are wrong! Paradoxes are
> > signs of insufficient knowledge or false pretenses.
>
> Of course, but there is nothing paradoxical about different orders
> of infinity. Indeed, it is your attempts to try to formalize the
> contention that all infinities are the same size that is ACTUALLY
> paradoxical.
>
> >
> > Curt, I'm trying to _fix_ math, for myself and others.
>
> Feel no fret: you MUST, someday, succeed, at least in
> your own small way -- for society advances
> one funeral at a time.

Hi George,

I debate internally whether to reply to you. I respond better to
rational discourse than to barking. (Snare snare cymbal.) I read
your insults as jocular overfamiliarity, because I choose to be not
offended. Please don't call me stupid.

About the ordinals, and sets as ordinals and ordinals as sets, the
notion is that each set represents an ordinal, the predecessor of its
order type. Thus, multiple sets represent the same ordinal. In that
way then, the powerset is the successor and order type of any set,
being as well an ordinal. I coined the phrase "ornate ordinal" as
contrasted to some plain ordinal, eg x U {x}.

You loudly state that no set maps bijectively to its powerset. You
might consider sharing that volume with Holmes and the NFU crowd, for
which that is not always so, although they basically have "large" and
"small" sets.

Me, I just have that f(x)=x+1 maps all elements of x, the set itself,
to all elements of x+1, the set itself, for finite or infinite sets,
neatly compounding the notions of powerset, order type, and successor.
 Technically, where it's an infinite set and its powerset, then
infinity is dually represented as zero, the case of the finite set is
mechanically true in that way, but not for each element, unless you
don't have empty set being a subset of every set.

You say that there are no paradoxes with Cantorian theory, and that I
am supposed to show ZF inconsistent. It is uncertain whether that
will happen, I'm thinking about it. You mention Russell's paradox,
you probably didn't catch my recent twist on it validating my zero
vis-a-vis infinity dual representation. There would be Burali-Forti,
that's similar to what would be Cantor's paradox, that the set of all
sets would indeed be its own powerset. With the above notion about
ordinals, they are combined into one single issue, where Ord is less
than nothing, and it itself is its own successor, which would again
perhaps be the empty set, which is the opposite and same as Ord. You
might notice that that is the statement of the singular proper class
and exluded excluded middle. About Skolem, in this thread again you
see what there is about Skolem in ZF: backpedalling and handwaving,
insensate to the infinite character, nature, of two infinite sets.

If you think about zero, you're probably not the first person to ever
do that. What comes around goes around.

You read my posts to sci.math, you know I use excluded middle from
nothing, or conversely, everything, to get the opposite. In that
sense consider this synthesis from ur-element(s):

null, U, 1, U-1, 2, U-2, etcetera

and consider how that illustrates an enumeration of the positive and
negative integers:

0, -1, 1, 2, -2, ...

Alternately, perhaps it is as so:

null, 1, 2, 3, ...

from which is extracted the set of natural numbers:

0, 1, 2, 3, ...

Notice how it is remarkably similar to a machine integer from the
binary transistorized register of a modern computer, except it's the
infinite word width, and you don't know whether binary one and zero is
inverted, nor is that at issue.

The idea here is to very mechanically symbolically implement field
operations on the integers, and then map those to various underlying
representations of the ordinals, for uniquification as well as
intrinsic composition identifers, to implement the number system as
close to the machine(s) as possible.

So anyways, about the naturals and reals and Cantor's antidiagonal
argument, there is the above resolution about any infinite set and its
infinite powerset each being infinite, but more specifically in
analysis of the natural/unit equivalency function as the compositional
building block of metrified mappings to and from the naturals and the
unit interval of the reals, binary or even decimal expansions are not
capable of representing these non-standard reals, each a real number.

You mention the "sci.math FAQ", the frequently asked questions, a
guideline of regularly encountered material. I recommend the Math
Forum's "Dr. Math", because they tend to discuss mathematics. I do
have to give .999... and 1 some credit, and offer resolutions about it
based upon these non-standard reals that .999... is a confusing
statement of 1 - iota, and that .999... equals 1, and 1 - iota does
not equal 1, because there is no expansion for 1 - iota, and between
zero and one there are only real numbers.

Besides that, though, maybe even a hundred people have, on their own
volition, registered disagreements with transfinite cardinals directly
to sci.math as it is the premier open forum for mathematical
discussion with the widest readership on the Internet with open
fishbowl type group critique. They vary in sophistication. As
objectively as possible, I have bias towards my own theory.

Mathematics shouldn't have any contentious issues, philosophy,
perhaps, mathematics, no. In mathematics, there are cut-and-dried
true-or-false answers, and various unknowns in terms of conjectures
with unproven truth values that are each decidable.

Warm regards,

Ross A Finlayson

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