Re: Cantor's diagonal proof wrong?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/16/04 

Date: 16 Nov 2004 11:15:57 -0800

curt@kcwc.com (Curt Welch) wrote in message news:<20041114152924.862$6K@newsreader.com>...
> raf@tiki-lounge.com (Ross A. Finlayson) wrote:
>
> > Curt, what's the point, man?
>
> I've addressed that to some extend in a previous post now. My interst
> comes from my exploring the ideas of AI. I'm not trying to "fix" math, I'm
> trying to understand what has happened.
>
> My interest is in understanding the relationship between what we can do
> with language, and what exists in the physical world. My interest is to
> try and understand if there is a clear point where we violate some
> important principle and end up describing something with language, that can
> never exist in our universe.
>
> I can make up a story about a blue book on my desk. There is no blue book
> on my desk, so I've just used language to describe something that does not
> exist. However, just because it does not happen to exist does not mean it
> is impossible for it to exist. I can describe a blue book on my desk which
> is talking to me. That's something we see in the cartoons all the time.
> And as far as we all know, such a thing does not exist in real life
> anywhere in the universe. But it could exist for all we know.
>
> But, is there some way to use language where we cross over from
> very-unlikely, to flat out impossible? I think there might be. I'm trying
> to understand if that point exists and how to describe it.
>
> I'm trying to understand if some fields of math might have wondered off
> into the "flat out impossible" land. And if they have, what it means for
> those fields of reason and how they relate to the fields of reason which
> have not left the land of the possible.
>
> > Anyways, Curt, some people are very attached to their notions ...
>
> That's key. People use language to justify what they believe. They seldom
> if ever, really understand why they believe what they believe, yet, if they
> can construct elaborate language to justify it, it makes them feel good, so
> they do it. We all work this way. And that's part of the danger. The
> language we use to justify everything exists simply because it makes us
> feel good. Separating truth, from "good feelings" is much harder to do
> than most people understand because in the end, none of cares as much about
> truth as we do about feeling good. (but that's another argument for another
> group).
>
> You used a lot of language about math in your post which I do not
> understand. I have a lot of work to do before I could discuss those issues
> with you. But I did find your post interesting.

Hi,

Yeah, Skolem is useful for those who don't care to deny that two
infinite sets are each infinite, if only as an example to give to
others, besides peer-reviewed papers of dubious validity on ArXiv, for
example, repository of much of cutting edge modern mathematical
physics.

While that is so, Cantor's results, as mathematical results, demand
evaluation.

In terms of the naturals and reals, any mapping of the naturals to and
from the reals would have to have characteristics thus that Cantor's,
and the few other quite similar results, about the naturals and reals,
do not hold, as each of the integers and reals are infinite.

Infinite: in-finite, not finite, neverending, without cessation,
there's always one more.

Casual examination leads to reevaluation of the definitions of the
reals, because standard definitions of the real numbers are inadequate
to express them. That leads to formal expressions.

Besides the natural and real numbers, there is the consideration of
the set, and its powerset. One reason set theory is deemed suitable
for the expression of what is considered the foundations of
mathematics is that it is so simple. Where infinite sets are
equivalent, and the naive interpretation of the infinite powerset
leads to what would be a paradox, or contradiction, which is not
allowed, then that leads to basically dual representation, an
enlightened interpretation.

You might have heard of "proper classes", where someone says "in ZF,
the set of all sets does not exist, there is only the proper class of
all sets." Under some naive and correct definitions of the proper
class, there can only be one or none of them. The empty set is the
proper class or ur-element, as is infinity. It's a _set_ theory,
founded on nothing and everything.

Once you have an infinity, you can do a lot of stuff with it. All the
infinite ordinals can be considered as ordinals, with Ord, the order
type of all ordinals, being that original infinity, one infinite value
among all the infinitely many finite positive integers, equal to zero.
 Maybe Kronecker had a point, there are only integers, all the rest is
the work of man, although that's just a quote that survived.
Kronecker was also probably abrasive and unfriendly, at least to
Cantor, they clashed, whereas I like everyone personally.

In a _very simple_ set theory, each ordinal is a set, and each set is
an ordinal. The powerset is just the order type, which is just the
successor. The function mapping ordinal to successor is f(x)=x+1, and
as well the root set of an ornate ordinal is the predecessor.

Being able to rationalize that Cantor's strong and subtle mathematical
results about infinity do not prevent me from considering bijections
between any two infinite sets, and to responsibly forget about
cardinal numbers, then the point is to immediately think of as many
useful things as possible that people, in particular creative
forward-thinking mathematicians, can not or would not consider because
they are bound by their orthodoxy, because when the dam breaks, they
won't be washed away, and I have a head start in developing
mathematical theories supplanting them. It's a wager, as it were.

One thing I named that is just ancient as hell is the natural/unit
equivalency function. I'm not a mathematical historian, I am quite
confident the construction is broadly considered, for example by
Banach and Tarski, in so few words.

Hey, people besides Curt, why is it that the only thing that anybody
on sci.math seriously argues _against_ is the diagonal theorem? Where
there's smoke, there's often fire. 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. 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.

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.

Curt, I'm trying to _fix_ math, for myself and others. Some people
are very attached to their notions.

Ahem. You've been fooled by Cantor. That's OK, he was fooled too.

Thank you for reading my post, I encourage you to read others about
mathematics and the foundations of mathematics, I've posted thousands
of them.

So, you can have a post-Cantorian set theory where infinite sets are
equivalent.

Warm regards,

Ross F.

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