Re: Cantor's diagonal proof wrong?

will.harris_at_gmail.com
Date: 12/17/04 

Date: 16 Dec 2004 21:17:04 -0800

Ross A. Finlayson wrote:
> 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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