Re: Does the number of nines increase?

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."