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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Mon, 8 Jul 2024 01:28:10 +0200
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Am 07.07.2024 um 22:19 schrieb Chris M. Thomasson:
> On 7/5/2024 6:26 AM, Moebius wrote:

>> Still, the rational numbers are countable! (Not enough "infinite 
>> infinities embedded in them"!)
> 
> [...] How many embedded infinite infinities are 
> "needed" _before_ it can be deemed uncountable? 

As much as are needed?

> Say between 0 and 1.

Well, the rationals won't do the job, but the reals can.

> There seems to be an infinite number of rationals that can fill in the 
> "gap", so to speak.

Sure, but not enough, though.

> 0 + (1/8 + 1/8 + 1/4 + 1/2) = 1
> 0 + (1/16 + 1/16 + 1/8 + 1/4 + 1/2) = 1
> 0 + (1/16 + 1/16 + 1/8 + 2/3 + 1/3 - 1/4) = 1
> 
> These are all rational, right?

Sure.

One of my professors once tried to express this state of affairs the 
following way: "There are (in a certain sense) much more real numbers 
than rational numbers."