Path: news.eternal-september.org!eternal-september.org!feeder3.eternal-september.org!news.szaf.org!news.karotte.org!news.space.net!news.muc.de!.POSTED.news.muc.de!not-for-mail
From: Alan Mackenzie <acm@muc.de>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Wed, 9 Apr 2025 16:17:37 -0000 (UTC)
Organization: muc.de e.V.
Message-ID: <vt66j1$10br$1@news.muc.de>
References: <vt3dg5$1qj4p$1@dont-email.me> <4DcJP.430046$dBr6.129536@fx04.ams4> <vt3sc2$1qj4p$2@dont-email.me> <UwfJP.698937$Kb9a.268084@fx16.ams4> <875xjes23l.fsf@nosuchdomain.example.com> <G7vJP.2125082$gHk7.1746361@fx17.ams4> <vt5v7a$2kqi$1@news.muc.de> <wqvJP.2103732$nb1.187919@fx01.ams4>
Injection-Date: Wed, 9 Apr 2025 16:17:37 -0000 (UTC)
Injection-Info: news.muc.de; posting-host="news.muc.de:2001:608:1000::2";
	logging-data="33147"; mail-complaints-to="news-admin@muc.de"
User-Agent: tin/2.6.4-20241224 ("Helmsdale") (FreeBSD/14.2-RELEASE-p1 (amd64))

Mr Flibble <flibble@red-dwarf.jmc.corp> wrote:
> On Wed, 09 Apr 2025 14:11:54 +0000, Alan Mackenzie wrote:

>> Mr Flibble <flibble@red-dwarf.jmc.corp> wrote:
>>> On Tue, 08 Apr 2025 15:46:54 -0700, Keith Thompson wrote:

>> [ .... ]

>>>> If you're interested in learning more, search for "surreal numbers" or
>>>> "hyperreal numbers".  If you're not, don't.

>>> Surreal numbers are bullshit as they don't actually exist, logically
>>> (as I have show).  Bullshit can be internally consistent with itself.

>> What exactly do you mean by a mathematical entity "not existing"?  What
>> is your test which partitions such entities into "existing" and
>> "non-existing"?

>>> /Flibble

> Simple: things that make no logical sense don't exist: ....

Surreal numbers do make logical sense.  They form an ordered field which
has the real numbers as a subfield.

> .... logically a real number always has a number smaller than it ....

Every stricly positive surreal number has a number smaller than it, too.

> .... so trying to put a "surreal" infinitesimal on the same number line
> as a "real" makes no logical sense: in fact I will go as far to say
> that it is a category error.

The surreal number line is not the real number line, so trying to put a
surreal on the latter indeed makes no sense.  It might even constitute a
category error, as you suggest.

That, however, has no bearing on the existence of surreal numbers.  They
don't create inconsistencies, hence do exist, and have been studied
intensively.

> /Flibble

-- 
Alan Mackenzie (Nuremberg, Germany).