The set of real numbers most certainly exists, it is defined as the unique such set (up to an isomorphism) that satisfies the following properties:

It is at the same time a commutative additive group and a commutative multiplicative group such that the two group neutrals aren't the same. Furthermore the multiplication distributes over addition.

There exists a total order relation such that the addition distributes over the order relation and multiplication agrees with the order relation (if (0 leq x) and (0 leq y) then (0 leq x*y)).

Another axiom is required, which comes in many forms (all of which are equivallent) so take your pick, for example the supremum axiom https://en.wikipedia.org/wiki/Least-upper-bound_property

You can read about any of this in any texbook of real analysis (for example Spivak or baby Rudin).

If the axiomatic formulation of real numbers doesn't appeal to you, there is another way to define real numbers, namely to contruct them from natural numbers (and you seem convinced that natural numbers indeed exist). You can read about that in the appendix to chapter 1 of baby Rudin (Principles of Mathematical Analysis by Walter Rudin).

By no means am I an expert in this stuff, but don't you need the axiom of choice (or maybe something just a little bit weaker) to construct the reals?

I don't think it's fair to say the reals 'most certainly exist' without being misleading to a layman. They exist given some axioms that are used overwhelmingly often in mathematics, but you can still do some interesting stuff without those axioms, or with their negation.

You're correct. There is one object in all of mathematics that isn't defined (you've got to start from somewhere), it is just assumed that humans implicitly understand this concept, and it's the concept of a set (denoting a collection of objects).

Along with it, certain properties of this object are assumed (among them the ability to choose an element of a non empty set - this is typically called the axiom of choice). For a full list you can take a look at http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

But the axiom of choice is logically independent of the other zermelo frankel axioms. You can assume zermelo franked plus choice, and get the standard framework (ZFC) that let's you build the reals.

But you can also assume ZF plus the negation of the axiom of choice, and get a system that is consistent if and only if ZF is consistent. It's not clear (to me, at least) that this other system will let you build the real numbers.

Clearly real numbers exist. I just don't believe you can have the set of every single real (they are unlimited).

Sets of numbers aren't physical things. Physical things can't be infinite, but I don't see why you think a set can't. By this thinking, not even the natural numbers can exist, since you can't point to them either.

You may enjoy:

https://en.wikipedia.org/wiki/Ultrafinitism

...and "Set Theory, Should You Believe?"

https://web.archive.org/web/20110616020815/http://web.maths....

The universe may be infinite.

Natural numbers are easy to write: 1, 2, 3

You can even write a set of them: {1, 2, 3, 4}

You cannot write the set of real numbers.

{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)

The set of real numbers is as big as the set of functions from the naturals to themselves. In fact, that's a pretty convenient way of modeling the real numbers sometimes: as systematic names for functions on natural numbers.