SPQR
Posts:
411
Registered:
8/12/11
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Re: Who's up for a friendly round of debating CANTORS PROOF?
Posted:
Aug 21, 2011 3:24 PM
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In article
<495dfe77-4526-4181-a77e-d167d4bdac25@u20g2000yqj.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 20 Aug., 23:38, Jim Burns <burns...@osu.edu> wrote:
>
> > Start with a listing of real numbers, ANY listing of real numbers,
> > complete, incomplete, whatever. Use the listing to construct a
> > real number that is NOT the first entry of the listing, NOT the
> > second entry, NOT the third entry, and so on. ("And so on" is
> > critical.) In fact, you can construct a real number that is NOT ANY
> > of the entries in the list. It is not in the list. Therefore,
> > THAT listing is NOT complete. What have we assumed about the listing?
> > ONLY that it is a listing of real numbers.
>
> No. In the first place we have assumed that Forall n in |N is a
> sensible quantifier application.
> Forall n in |N means that there is no natural number missing.
> On the other hand we have:
> For all n in |N: The set {1, 2, 3, ..., n} is not |N.
> On the contrary:
> For all n in |N: there are infinitely many natural numbers missing in
> {1, 2, 3, ..., n}.
> For all n in |N: there remain infinitely many lines in Cantor's list
> that are not checked.
Since we have a general rule for checking which checks all lines
simultaneously, which lines do you claim are exempt from that general
rule.
Example: "For all n in |N, n + 1 is greater than n" is a general rule
which is true for all n in |N, so has no exceptions at all.
So when a rule wish works correctly for every line is applied, for which
lines does WM claim it does NOT apply?
>
> > Therefore, we have proven
> > that ANY listing of real numbers is an INCOMPLETE listing of real
> > numbers.
>
> More than that: Any listing of natural numbers is incomplete. It is
> impossible to check it completely.
How is "f:|N --> |N : n |--> n" incomplete? It certainly "lists" every
natural number.
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