Re: More complex numbers than reals?

Am 09.07.2024 um 22:27 schrieb sobriquet:

How do you define sets exactly?

Actually, we don't _define_ the concept of /set/ by a "proper definition".

Is there a specific set that corresponds to sqrt(2)?

Well, rather a sequence (which is a certain kind of set in the context of set theory):
(1, 1.4, 1.41, 1.414, ...)

Does this set have an infinite number of elements analogous to the sqrt(2) having an infinite decimal expansion?

Yes. See above. This sequence (called an /infinite sequence/) has infinitely many terms.

It seems that the existence of something like sqrt(2) is already rather dubious.

Oh, really?
If you say so.
So in your "math" there is no /number/ x such that x^2 = 2.
Ok, if you can live with(out) that, fine.

In reality, things are finite and space and time might also be finite (composed of atoms of space and time that can't be subdivided with
the parts retaining their original spatial and temporal properties).

Yes, they could.

So if the concept of irrational numbers like sqrt(2) [etc.]

Hint (1): You won't find numbers like sqrt(2) IN (PHYSICAL) REALITY.
Hint (2): You won't find numbers like 1, 2, 3 there neither/either (?).