Re: Cantor's diagonal proof wrong?

From: George Greene (greeneg_at_cs.unc.edu)
Date: 11/17/04 

Date: 16 Nov 2004 17:14:57 -0800

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.

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