> It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat.

If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.

> If you accept countably infinite rationals then you also accept Cauchy Sequences, no?

Absolutely.

When most people think of reals, they are thinking constructible reals. Which are countably infinite in number.

If there is an equation (known or in principle), they are constructible.

If there is an algorithm (known or in principle), they are constructible.

Limits, continuous fractions, all those are constructible.

From a mathematics perspective, I think it's a loss that the distinction between countable/constructible numbers vs. uncountable/un-constructible is completely blurred under one name "reals" early in everyone's math education. Even though the difference is significant when reasoning about information, the relationship between math and physics, math and computation, etc.

And about infinites. Most of us famously know that there are infinitely more reals than integers.

But how many people know that all well defined reals, constructible reals, calculable reals, and their equations, probably everything they imagine when they think of "real" numbers in practice, remain countably infinite. Exactly like the integers.

The set of constructible reals is the same size as the set of integers.