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-

>

> list beyond line n contains infinitely many rational numbers q_k that

>

> 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

>

> 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

>

> 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

>

> 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