Re: Cantor's diagonal proof wrong?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/14/04 

Date: 14 Nov 2004 11:04:28 -0800

Virgil <ITSnetNOTcom#virgil@COMCAST.com> wrote in message news:<ITSnetNOTcom#virgil-096A0C.00323214112004@comcast.dca.giganews.com>...
> In article <20041114013915.877$0a@newsreader.com>,
> curt@kcwc.com (Curt Welch) wrote:
>
> > Here's something all of you should have some fun with.
> >
> > Nath is not something I specialize in (and I don't read this group
> > normally), but I've been looking at a few things lately and I've decided
> > that some very big mistakes have been made in math because people started
> > playing around the concept of infinity without realizing the trouble they
> > were creating for themselves.
> >
> > When I was shown Cantor's diagonal proof that the number of reals was not
> > countable back in college, I thought it was a fascinating proof. It seemed
> > to uncover some great mystery about the nature of numbers that was not at
> > first obvious. It sounded very logical and I quickly embraced it as fact.
> >
> > Lately however, I've come to see things very differently. I now belief the
> > proof is totally bogus. And the huge body of work built on top of the
> > concept is likewise, totally bogus.
>
> AS the "diagonal " proof was Cantor's SECOND proof of the uncountability
> of the reals, and there have been several subsequent proofs, all of
> which are totally independent of the "diagonal" construction, it would
> not affect the validity of the theorem itself even if the "diagonal"
> proof were to be found flawed.
>
> For which reason, no sensible mathematician is the least worried that
> such a flaw would in any way weaken the validity of the theorem itself.

Hi,

I'm writing to belabor "the binary case is sufficient and necessary."
I'm reminded of my request about belaborment, which was about
communication and confusion issues, Virgil. Why do you think the
antidiagonal argument is flawed?

In the binary case, there is one specific anti-diagonal.

Consider an arbitrary base. Any method you use to generate some
antidiagonal will affect more than one location in the expansion as a
binary number. In that way, it might reset one of the previous
locations that would have been different, thus that the antidiagonal
would not be different at that location. That's an implication that a
number represented in a different base is a different number, and
stranger things are known to occur. That is perhaps just an artifact
of the algorithm.

That's similar to the argument that any number is representable in any
radix (base). The point is being that if there is some list, to
generate the list in a base greater than three, where three is as well
shown useless as a base to definitely generate an antidiagonal, and
construct an antidiagonal in some way that it is not rational so it
couldn't be dually represented.

Add a leading zero to each element of the list, then only in a
specific case is the antidiagonal an element in the range.

You refer to other arguments about the naturals and the reals, so do
I. With regards to the nested intervals, they are not constructed,
with EF. Then again, my line of reasoning easily uses what you would
not term standard real numbers.

The rationals are dense in the real numbers.

Curt, you might want to learn about Skolem. Skolem extended the work
of Loewenheim to show that everything is countable. People handwave
about that and they're quite nonsensical in their ludicrous nonsense,
because the extensions are no different than the set. What that means
is they say that they have a set of integers that maps to a powerset
of integers, but in a receding slippery slope type of way that still
claims the opposite true. That's why they call that quandary Skolem's
paradox.

http://groups.google.com/groups?q=Skolem+Cantor

If you accept that the powerset result does not always hold true,
then, both Skolem's and Cantor's "paradoxes" dissolve, where Cantor's
is that a set of all sets would be its own powerset, and would map to
itself with the identity function. Without transfinite cardinals, for
everyone, measure theory needs some few new foundations, or rather,
just rephrased foundations, with perhaps some meaningful results, and,
that is about it, and all of transfinite cardinal mathematics is its
own little subfield where you axiomatize that so, just so all the work
put into transfinite cardinals was not a total waste of time, like a
pickled three-headed sheep.

Curt, what's the point, man? Do you want to map the reals to the
integers? What good is mapping the reals to the integers? Do you
think calculus is easier to understand if dx is a llittle
infinitesimal coefficient and when you sum the product of the function
and dx over the range that you get the integral? Even if that was too
slippery for general use, the limit being a safety feature of sorts,
and all the calculus was done using limit, wouldn't that be better?

Me, I was just offended that somebody claimed infinite sets weren't
equivalent. Now I feel better about it, because I've proved a few
things about that to people.

Do Zeno's paradoxes prevent Achilles from catching the tortoise? No,
they don't. Does Skolem's paradox prevent there being uncountable
sets? It does. You've probably heard of the "paradoxical" barber,
there are no paradoxes and so that barber does not exist anywhere,
because everybody in that town is shaved by the barber unless (if and
only if they don't) they shave themselves, everybody shaves, nobody
ever leaves town, and the well-meaning barber, who as an expert
probably shaves himself, also is the barber shaving himself. So, the
barber shaves himself and anybody else who doesn't shave themself.

Take two infinite sets. If there is a way that for each you can
select an element of each set and remove it from that set, do that.
That's a terse constructive proof that infinite sets are equivalent.

Cantor's results have meaning, they in a way force certain conclusions
about the nature of binary logic, because of that one element that is
unmapped, call it the antidiagonal or something, infinity rolls right
back over to zero like an odometer.

That gets into that any set X is an ordinal, and that the order type,
and successor, and X+1, and the powerset, are all the same thing.

When you're talking about mapping the naturals to the reals, there's
probably actually some useful formulas or "functions" that be used to
derive mathematical results that are not otherwise immediately
apparent. Here's a mapping between the natural numbers (0, 1, 2, ...,
non-negative integers) and the unit interval of the reals ( R[0,1],
every real number between zero and one inclusive): the natural/unit
equivalency function, EF. It's simple, order the reals from least to
greatest or greatest to least, and then map zero from the integer to
least or greatest, and then, in order, pair elements of those sets.
The binary antidiagonal does not exist on the range or is dually
represented, or you can add leading zeroes, and non-standard real
numbers, which are very much real numbers, are used thus that results
about mapping the naturals and reals do not apply. So anyways,
integrate EF and the result is equal to one, where you might think it
would be equal to one half, because you'd figure it would be just like
f(x)=x from zero to one.

That has to do with how points on the real number line are defined in
terms of preceding and following points on that same line, and that
points on a continuous line are in a sense one-sided, where that side
is in the direction of the ray's passage on that line, as the reals
are ordered thus that for two different real numbers one is lesser and
one is greater, or oppositely one is greater and one is lesser. When
the number is by itself then it has two sides and twice the weight,
because two different straight lines can pass through it.

You may as well consider a different method for sweeping through those
points, such as a spiral of sorts or alternatively taking the next
indefinite real element on the lesser and greater side. Again, that
leads to models of non-standard reals, which are real numbers.

Anyways, Curt, some people are very attached to their notions of
cardinality and the uncountable, they think it's very sophisticated
and urbane. A lot of work has been done based upon the simple notion
that f(x) = x+1 doesn't equal x+1. Most don't give a damn either way.

You can say that half of the integers are even, and that half of the
integers are positive, and that a given fraction of the integers are
primes or perfect squares, without the necessity of the transfinite
cardinals. As well, it is shown that a proper subset of a set has
less elements than the superset. There are more rationals than
integers, and more reals than irrationals or rationals. A powerset
has more elements than the set, in a sense, that's not the problem.

Curt, 1+1=2, and 2+2=4. Can't you leave the Cantorians their paradise
and well enough alone? Biblically, Adam and Eve were cast from
Paradise after they partook of the tree of knowledge. If they hadn't,
they'd still be there and that would be the end of the story.

Warm regards,

Ross F.

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