Date: Jun 25, 2013 5:27 PM
Author: Tucsondrew@me.com
Subject: Re: Joel David Hamkins on definable real numbers in analysis

On Tuesday, June 25, 2013 1:43:36 PM UTC-7, muec...@rz.fh-augsburg.de wrote:
> On Tuesday, 25 June 2013 17:07:36 UTC+2, dull...@sprynet.com wrote:
>

> >> There are not uncountably many reals, neither in the natural order nor in the well-order.
>
>
>
>
>

> > That's very funny.
>
>
>
> That's easily provable:
>
>
>
> Consider a Cantor-list that contains a complete sequence (q_k) of all
>
> rational numbers q_k. The first n digits of the anti-diagonal d are
>
> d_1, d_2, d_3, ..., d_n. It can be shown *for every n* that the Cantor-
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> list beyond line n contains infinitely many rational numbers q_k that
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> have the same sequence of first n digits as the anti-diagonal d.
>
>
>
> Proof: There are infinitely many rationals q_k with this property. All
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> are in the list by definition. At most n of them are in the first n
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> lines of the list. Infinitely many must exist in the remaining part of
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> the list. So we have obtained:
>
>
>
> For all n exists k:
>
> d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., q_kn.
>
> This theorem it is not less important than Cantor's theorem: For all
>
> k: d =/= q_k.
>
>
>
> Both theorems contradict each other with the result that finished
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> infinity as presumed for transfinite set theory is not a valid
>
> mathematical notion.
>


The statement, "If n e N, then FIS_n(d) e S."
does NOT imply that "d e S".

>
> Regards, WM


ZG