{1, 2, 3, 4} is also a set of real numbers. Why is it a valid existence proof of the naturals, but not the reals?

Possible linguistic pedantry; he used "a" set of natural numbers, but "the" set of real numbers. It's equally difficult to represent the set of natural numbers.

It's not pedantry, it's just what the words mean.

To the question: I'm only saying that the "set of real numbers" and the "set of natural numbers" don't seem to exist.

Despite numerous downvotes, no one has yet produced them here (or even a link to them).

I can't produce the set of all humans on Earth either, but that doesn't mean they don't exist.

Both real and natural numbers can be reasoned about despite their infinite size (and we even know that e.g. there must be "more" real numbers than natural numbers, even though both sets are infinite)

The set of all humans is produceable, you simply bound by the planet, or solar system.

There are no bounds in the universe that can contain all the real or natural numbers.

If real numbers exist or not is irrelevant. The question is if we can solve some problems by assuming one way or another.

here is a representation of the real numbers: ℝ

and here is one of the natural numbers: ℕ

How can I select an arbitrary, or even random, element from either?

Who said that you could?

Zermelo?

Axiom of choice says I should be able to select an element from the set of real numbers (assuming it exists; but, I believe the assumption to be counterfactual).

I agree. Even though the axiom of choice is independent of ZF, I don't think it is self-evident axiom. It actually seems pretty counter-intuitive if you believe in infinite-precision real numbers that have an infinite amount of information and can't be compressed. I have more to say on this in "Digital Physics" (the movie). -Khatchig

According to Wikipedia, Zermelo formulated the axiom of choice.

I think it makes sense though; if I (claim to) have a thing, I should be able to choose/pick/select/point-to it (seems to be almost[?] tautological).

"Say you are playing a game that needs you to pick a real number. If you choose a computable real number, you lose the game. If you choose a real number that is not computable, which the majority real numbers are, then you win. You can imagine yourself choosing a non-computable real number, and winning the game, if you build in the axiom of choice. But in the real world version of the game you will never have enough time or space to non-ambiguously specify this infinitely precise real number which has an infinite amount of non-compressible information (see Kolmogorov complexity)."-Khatchig, from "Digital Physics" (the movie)