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Topic: Joel David Hamkins on definable real numbers in analysis
Replies: 98   Last Post: Jun 30, 2013 3:22 PM

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mueckenh@rz.fh-augsburg.de

Posts: 14,649
Registered: 1/29/05
Re: Joel David Hamkins on definable real numbers in analysis
Posted: Jun 25, 2013 4:43 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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
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.

Regards, WM


Date Subject Author
6/18/13
Read Joel David Hamkins on definable real numbers in analysis
fom
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David Petry
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
fom
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Peter Percival
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/21/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/21/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/21/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/24/13
Read Re: WM screws up the notion of a limit!
Virgil
6/25/13
Read Re: WM screws up the notion of a limit!
Virgil
6/25/13
Read Re: WM screws up the notion of a limit!
Virgil
6/26/13
Read Re: WM screws up the notion of a limit!
Virgil
6/26/13
Read Re: WM screws up the notion of a limit!
fom
6/26/13
Read Re: WM screws up the notion of a limit!
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/22/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/23/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/24/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/25/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Ralf Bader
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
David C. Ullrich
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/27/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/27/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/26/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/27/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/27/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Tucsondrew@me.com
6/29/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/29/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/30/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/29/13
Read Re: Joel David Hamkins on definable real numbers in analysis
mueckenh@rz.fh-augsburg.de
6/29/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/28/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil
6/19/13
Read Re: Joel David Hamkins on definable real numbers in analysis
Virgil

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