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From: Moebius <invalid@example.invalid>
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Subject: Re: More complex numbers than reals?
Date: Wed, 10 Jul 2024 23:39:59 +0200
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Am 10.07.2024 um 23:23 schrieb Chris M. Thomasson:

> Well, it missed an infinite number of reals between 1 and 2. So, the 
> reals are denser than the naturals. Fair enough? 

In a certain sense, yes.

On the other hand, the rational numbers are "denser than the naturals" 
too (in this sense). Still, the set of rational numbers has the same 
"size" as the set of natural numbers.

Man, try to comprehend that once and for all. :-P

(Mückenheim never succeeded concerning this point. Please try to avoid 
this rabbit hole.)

> It just seems to have "more", so to speak.

Exactly!

The real numbers actually do, while the rational numbers don't.

> Perhaps using the word "more" is just wrong.

EXACTLY. Well, it's certainly "misleading" when dealing with INFIITE SETS!

That's the very point Mückenheim can't comprehend.

> However, the density of an infinity makes sense to me. Not sure why, it 
> just does...

It does.

Hint: Between any two (different) rational or real numbers there's 
another rational / real number.

That's not the case for natural numbers. They are not "dense".

> The set of evens and odds has an [countably] infinite number of elements. Just like 
> the set of naturals.

Exactly.

Thats why we accept (in set theory) that these sets have "the same 
size". (Of course, this is by convention. But so what?)