Re: Cantor Diagonal Proof

Mr Flibble <flibble@red-dwarf.jmc.corp> wrote:

On Wed, 09 Apr 2025 16:17:37 +0000, Alan Mackenzie wrote:

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

The category error I identified runs contrary to your claim that the reals
are a sub-field of the surreals ....

It's not my claim.  It's an established mathematical fact.

.... as that would suggest that reals and surreals can exist on the
same number line as a real is-a surreal which is logically unsound for
the reason I have already given.

You're mistaken.  You appear not to understand the implications and
meaning of the word "is" in that last sentence.  In this subthread, you
haven't given any reason for your assertion.  And even if you had, you'd
still be mistaken.

/Flibble

/Flibble

--
Alan Mackenzie (Nuremberg, Germany).