Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 12/21/04 

Date: Mon, 20 Dec 2004 17:44:09 -0800

> From: cbrown@cbrownsystems.com (Chas Brown)
> By "a real number" r in R, I mean here a function (algorithm)
> r:N->{0,1} which we can understand as the binary decimal expansion of
> "the real number r"; i.e., with r(i) being the "ith" digit in r's
> binary expansion.

That's a bit sloppy. You seem to be defining "a real number" r and "the
real number r" to be two different things, the former being a binary
decimal expansion of the latter. Note that the two sets don't match up,
because there can be more than one "a real number" r which correspond
to a single "the real number r". For example, "the real number r" can
be 5, and corresponding to that "a real number" r can be either of
these:
"101.00000000..."
"100.11111111..."
(... in that notation means forever same binary digits as just preceding)
Accordingly it isn't correct to say "the binary decimal expansion of ...".

> We _define_ the following term: the statement "two real numbers r and
> s are equal" is equivalent to saying "for all naturals m, r(m) =
> s(m)". It follows that, if there exists a natural m with r(m) not
> equal to s(m) then r does not "equal" s.

That's most definitely faulty, because the two binary decimal
expansions of 5 are not equal by that definition.
Cantor's theorem is about real numbers, not binary decimal expansions
of real numbers. Binary/decimal/whatever expansions of real numbers are
used as a means to an end, but that's not what the theorem is supposed
to prove. Your overall logic is correct, but your details are wrong, so
you've set up a strawman for the trolls to attack as if your false
proof represented Cantor's correct proof.