Re: Cantor's diagonal proof wrong?

From: robert j. kolker (nowhere_at_nowhere.net)
Date: 11/17/04 

Date: Wed, 17 Nov 2004 08:51:38 -0500

Curt Welch wrote:
>
> I don't actually accept the existence of infinite length anything including
> irrational numbers. I only accept that the descriptions for the generation
> of infinite length irrational numbers exist. That is, we have a
> description for what the square root of two is. We don't actually have,
> the full, square root of two, written down anywhere.

The length corresponding to the square root of two is easility
constructble and laid out before your eyes. Construct a unit square and
draw a line segement between a pair of diagonally opposed corners of the
squre. There it is, the square root of two. Right in front of your eyes.

There is also a finite algorithm which will yield the n-th digit of the
decimal expansion for the square root of two.

>
> Likewise, we don't have the full list of natural numbers written down
> anywhere.

Not enough paper or disk space. But we can generate as many as we need.
There is no limit.

>
> The "mirroring the digits at the decimal point" algrorithm, is a
> description for generating real values, that will generate the same
> numbers, which any of the descriptions anyone can provide for any
> irrational (converted to decimal), will also generate.
>
> So to me, my description will, through obvious inspection, produce a one to
> one mapping to any decimal description of an irrational numbers that I know
> about.
>
> But, what I do accept, is this is not how mathematicians choose to define
> or think about these things. So, before I can understand what they have
> done, and why they think it's valid, I have to learn their language.
> There's a huge body of work formalizing how they do think, which I have not
> yet mastered.

And you probably never will. Yoda says: Hold not your breath young Curt,
else purple turn you will.
>
> What remains a mystery to me is that I still don't see how any formal
> system could be both consistent, and produce the results which Cantor's
> proof produces.

That is admirably correct. You don't see.

Why don't you try a line of thought where you CAN see (i.e. understans).

Bob Kolker

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