[CJefferson]

I don't agree, but I agree it's an interesting discussion to have.

When is the set of all possible subsets of natural numbers worth considering more than the set of all sets which don't contain themselves (which gets us Russell's paradox of course), once we start building infinite sets non-constructively?

The naturals to me are a clearly separate category, as I can easily write down an algorithm which will make any natural number given enough time. But then, I'm a constructionist at heart, so I would like that.

[kcexn]

You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals.

Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.

[CJefferson]

Except we can only describe those infinite series for a countably infinite number of the reals, so there are all these reals expressed by infinite series we don’t have any way to describe. Why do we need those ones? (To be clear, I realise this isn’t the current standard opinion of most mathematicians, I choose to be annoying).

[kcexn]

It's been a while since I did abstract algebra, but I'm pretty sure that once you have the additive and multiplicative identities, the rest of the reals can be generated. Which is still a constructive process.

Regardless, the existence of the real numbers is not a matter of need. Their existence is a consequence of how mathematics is defined. Over-simplified, it's a case of if addition and multiplication work, then the real numbers must exist.

Usually, maths doesn't require us to overthink about anything metaphysical. Things either are or they aren't, the problem-solving approach taken to demonstrate a result one way or the other is the fascinating part.