[osoba]

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).

[soVeryTired]

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.

[osoba]

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

[soVeryTired]

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.

[jsprogrammer]

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