Re: The consise Cantor's proof ?

rupertmccallum_at_yahoo.com
Date: 12/16/04 

Date: 16 Dec 2004 05:35:41 -0800

|-|erc wrote:
> I just checked google, I'm missing Rupert and Charlie's responses,
thought
> the post was left unreplied to. Yes its one for DCProof I think, who
can
> make the shortest formal proof. Not that I accept Er e R outside of
the Eb e N.
>
> Here's my new slant/
>
> every rational you can represent lines up with some natural on a
single list.
> every real you can represent lines up with some natural on a single
list.
>
> no contradicitons, no need for hyperinfinity.
>
> -----------------
>
> Each row is a real from 0 to 0.999..
> eg. r3(4) = UTM(3, 4) mod 10, the 4th digit of the 3rd real.
>
> To represent the anti-diagonal of UTM(x, y) mod 10 (ignoring halt
issues)
>
> you can only write such as r(x) = UTM(x, x)+1 mod 10 (mod 9 removes
the 0.999.. issue)
>
> UTM is a program itself TM-u for some natural number u.
>

This is incorrect.

> Therefore
> UTM(x,x) + 1 mod 10 = TM-u(x,x) + 1 mod 10 = UTM(u, (x,x)) +1 mod 10
>
> Let
> TM-v(d) = TM-u(d,d) - 1
>
> Then
> UTM(x,x) + 1 mod 10 = UTM(v, x)
>
> When x = v
> UTM(x,x) + 1 = UTM(x,x)
> CONTRADICTION
>
> Therefore there is no valid representation of a real UTM(x,x)+1 mod
10
>
> Herc
>
> --
> jab jab jab quote I'm a critical thinker
>
> "george" <greeneg@cs.unc.edu> wrote in
> > > Of course the free b in (r = !L(b)) puts doubt
> > > on such an algebraic derivation of Cantor's
> > > proof.
> >
> > Such an algebraic derivation AS WHAT?
> > Proofs ARE derivations. There is no such
> > thing as a derivation, ALGEBRAIC OR otherwise,
> > of a proof. There is rather a
> > derivation OF the THEOREM proved,
> > FROM the axioms and hypotheses, and
> > that derivation IS the proof.
> >
> > As somebody else has already pointed
> > out, "r=!L(b)" is not even a valid step
> > in your proof, let alone a reason why
> > somebody might find an algebraic
> > definition difficult. Your proof does
> > require one to make infinitely many
> > choices in going from
> >
> > > 4a. not(exists a, forall b, L(a,b) != L(b,b)
> > > (negation of 3)
> > to
> > > 4b. not(exists a, forall b, L(a,b) =!L(b,b)
> > > (! is some suitable digit change function)
> >
> > but even THAT infinity does not involve
> > anything suspect (like the axiom of choice),
> > because it is easy to stipulate a finitary
> > rule governing the choice, because there is
> > only a constant
> > finite number of different numerals for
> > DIGITS in these representations.
> > For maximum clarity it should just be 2,
> > in which case there is no choice because
> > there is only 1 digit-change function.
> > You are making infinitely many choices from
> > infinitely many pairs of shoes, not pairs of socks.
> >
> > The end result of this is that you finally
> > believe Cantor's theorem after all.
> > So why can't you just admit you were
> > stupid for not seeing it in the first place
> > instead of claiming (falsely) that you have
> > invented something insightful and concise
> > that might be of future value?
> >
> > The concise version of this proof is the
> > set-theoretical one showing that
> > any function has a subset of its domain
> > set has a subset of that set (an element
> > of its domain's powerset) that is NOT in its range.
> > Af[y(f)cDom(f) & Ab[~f(b)=y(f)]].
> >
> > That there really does exist such a y(f) for
> > every f really is a theorem.
> > You don't even have to arrange
> > the domain in a list. This whole problem
> > is much simpler and deeper than that and
> > the infinite-list version WORKS for reasons
> > having NOTHING to do with infinity OR
> > list-hood!
> >
> > Dave Seaman posted a concise proof
> > of this in ZF to the newsgroup years ago.
> >

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