Re: Cantor's diagonal proof wrong?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/28/04 

Date: 27 Nov 2004 21:22:50 -0800

Hi Bob, if that is your real name,

It's an infinite set, there's always one more.

Consider the natural/unit equivalency function. I make expansive
statements.

One representation of the binary expansions to represent a number is
by the integer modulus, that's the normal way, there are other ways to
encode integers in a binary string, for example Gray coding, a
"non-weighted reflected binary code." In Gray coding, an increment
changes one bit of the string. On many digital computers, the same
register is a given power of two number of bits is used to represent
the signed or unsigned machine integer quantity.

Is not it neat to think that half of the integers are non-negative and
the other half negative? Not including zero, if I tell you I've
selected an integer and deemed it positive or negative by flipping a
coin, is not that fair? Half of all the infinitely many integers are
even.

About which antidiagonals you might create, in binary there is only
one, and for EF it is dually represented or not on the range, and its
addition to the list doesn't change the list, because of dual
representation, because the pseudo-expansions of those n.s. real
numbers would have an antidiagonal with infinitely many repeating
terminating zeroes.

For the decimal case or b>3, there is the inductive impasse, for any
finite base there is a smaller interval thus represented, and then
thus through a harsh undecomposable composition there is thus a
trivial mapping to the rest of the real numbers. Reconsider the
immediate and deferred. This is again with these nonstandard
representations of all the real numbers. Can you not think
abstractly?

Again, as the integer part of a real number is a finite natural
number, a single leading zero can be added thus that the antidiagonal
is never on the range, inductively. It's nice to know that there
exist real numbers larger than zero.

Consider the expansion as representation in a finite "base", or radix
or integral modulus, in an infinite "base" the expansion for the
integer part is always one element.

For your inductive antidiagonal algorithm to generate a "list" of real
numbers, your list would have to satisfy the extensions of the nested
intervals style proofs. EF does, with its nonstandard definition of
the real number, and the antidiagonal algorithm of your choice
generates a value that is never within a finite interval.

Draw the line segment with the pencil, that involves continuous
application of the pencil to the paper with the infinitely precise and
hard marking pencil point and infinitely smooth markable paper. You
can photocopy your line with magnification, but if you magnify a list
of stippled points each applied individually, there would always be
gaps, the reals are not complete without the line, taking into account
the nature of the end- and non-endpoints on the line. There are only
points on the line.

Inductively, infinite sets are equivalent. That demands not denial
nor acceptance but revision and extension of these naively true
statements that contradict with other naively true statements.
Infinite sets are equivalent.

Skolemize: your model is countable. Hausdorff said: uncountable is
countable union of countable. The set of all sets, a set in some set
theories, would be its own powerset. In some theories, it's not its
own powerset twice. Are there any applications solely attributable to
the transfinite cardinals besides themselves? In some theories, the
powerset is order type is successor. The line is a set of points.

At the edge of theory all is nothing and nothing is all, and it's the
beginning of the theory. Stare into the abyss. It's a mirror.

I remember one time on PBS there was a mystery with a lady, she was a
member of the cult that stared at the Sun and went blind, she walked
into the open elevator shaft. She stopped at the bottom.

Concreteness is a goal. Borel and combinatorics can agree, they might
need some mediation, and clarification of definitions.

Ross

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