Re: Cantor's diagonal proof wrong?

From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 11/15/04 

Date: Mon, 15 Nov 2004 17:34:40 -0500

In <20041114122706.926$Pt@newsreader.com>, on 11/14/2004
   at 05:27 PM, curt@kcwc.com (Curt Welch) said:

>Why is it ok to write 0.111... but not ...11111 ?

It'sa OK to write either one, as long as you define what they mean and
don't assum e properties beyond those implied by their definitions. In
your case you are assuming that ...11111 is an integer, which is
patently false.

>It's just a name we use to talk about the
>real value which is 0.1111 repeating forever.

No. 0.111... is a name of limit n->oo 0.1 (n times), which happens to
be 1/9. You haven't defined ...11111 to be the name anything.

>And I can just as easily
>define the integer of 1 repeating forver.

That's not a definition.

>The only reason we do not do that is a matter of convention.

No, the only reason that we don't do it is that there is no sensible
mapping from such strings into the integers.

>It's not (so I claim) in violation of what integers are.

You can claim that 1+1=3 if you want, but that won't cause anybody to
take you seriously. Claims must be backed up with sound reasoning.

>If you start with 0, and continue to apply the +1 function to it,
>and ignore all the values you come up with which does not have all
>1's, you find you have the exact same type of defintion that gives
>you 1/9 when you generate a string of one's running to the right,
>instead of running to the left.

No. Again, you do not understand what an integer is or what a number
is.

>Also, all integers have an implied infinite string of 0's running to
>left

No. An integer is not a string of digits.

>(just like reals have am implied infinite string of 0's running to
>the left and right).

No. A real number is not a string of digits. There are partial
mappings from strings of digits to numbers.

>Just change the implied 0, to an implied 1,

That gives you a string of digits. It does not give you a number. 

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