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| From | wm <wolfgang.mueckenheim@tha.de> |
|---|---|
| Newsgroups | sci.logic, sci.math |
| Subject | Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) |
| Date | 2026-03-25 18:17 +0100 |
| Organization | tha |
| Message-ID | <10q15b8$hh2u$1@solani.org> (permalink) |
| References | (9 earlier) <10pr2g7$1gvg$1@dont-email.me> <10prmgr$dqm9$1@solani.org> <10pti86$sb40$1@dont-email.me> <10puonp$1araa$1@dont-email.me> <10pv0kr$1cmht$3@dont-email.me> |
Cross-posted to 2 groups.
Am 24.03.2026 um 22:44 schrieb Tristan Wibberley:
> On 24/03/2026 19:29, WM wrote:
>
>> (2) According to Cantor's definition of countable set the set of nodes
>> of the Binary Tree is countable.
>> (3) If we map every node onto a path, then the mapped set of paths is
>> countable.
>> (4) For every n ∈ ℕ: I map the nth node on a path containing this node.
>
> I'm not sure I understand your meaning as I'm expecting "to" instead of
> "on".
>
Maybe better.
>
>> (5) Therefore every npode is covered by this set of paths. There does
>> not exist a node which is not covered by these paths.
>> (6) From the root to every level L(k) the Binary Tree is completely
>> covered by this set of paths, for every k ∈ ℕ.
>> (7) These paths represent the real numbers between 0 and 1.
>
> What, all of them? Isn't it limited to some of the rationals being 1/2^m?
>
In fact the irrational numbers are limits only. They have no
representation by digits or bits. But usually it is claimed that the
Binary Tree conztains all real numbers of the unit interval. In
particular Cantor's diagonal argument concerns only digits. In fact it
would not apply to irrational let alone transcendental numbers. But that
argument would already show the contradiction.
>> (8) It is impossible to find a further real number between 0 and 1.
> ^^^^
> even if you changed that to "rational" I don't think it would be true
> and you provide no argument for the truth of (8).
Look at the short Binary Tree of only two levels L1 and L2:
N0
/ \
L1 N1 N2
/ \ / \
L2 N3 N4 N5 N6
...
The paths of the set P = {P(n) | n ∈ ℕ}, where node N(n) is mapped to
path P(n) fill the tree such that no further path can be distinguished.
Same holds down to every level Ln. My paths fill the complete Binary
Tree. Another path produced by the diagonal argument would have to
separate itself at some node. But there is no such node.
Regards, WM
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Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-03-24 21:44 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-24 19:31 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-24 19:32 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-24 20:07 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-25 14:04 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-04-04 20:54 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) wm <wolfgang.mueckenheim@tha.de> - 2026-03-25 18:17 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Alan Mackenzie <acm@muc.de> - 2026-03-25 17:55 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) WM <wolfgang.mueckenheim@tha.de> - 2026-03-25 21:18 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Alan Mackenzie <acm@muc.de> - 2026-03-25 21:17 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) WM <wolfgang.mueckenheim@tha.de> - 2026-03-25 22:53 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-25 15:08 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-26 10:28 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) wm <wolfgang.mueckenheim@tha.de> - 2026-03-26 17:33 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Alan Mackenzie <acm@muc.de> - 2026-03-26 17:34 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) WM <wolfgang.mueckenheim@tha.de> - 2026-03-26 21:08 +0100
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-27 09:21 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-03-26 12:22 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Alan Mackenzie <acm@muc.de> - 2026-03-26 12:40 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-03-26 13:09 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Alan Mackenzie <acm@muc.de> - 2026-03-26 13:40 +0000
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Moebius <invalid@example.invalid> - 2026-04-30 04:43 +0200
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-04-30 09:10 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Mikko <mikko.levanto@iki.fi> - 2026-05-01 11:55 +0300
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-01 09:47 -0700
Re: Criticism of a proof of a contradiction in set theory (Was: Re: AI understands where 99 % of mathematicians fail) wm <wolfgang.mueckenheim@tha.de> - 2026-05-01 15:20 +0200
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