TITLE: Understanding the philosophy behind the axiomatic definition of reals
QUESTION [7 upvotes]: I have just recently started to read a book about mathematics and have stumbled upon an axiomatic definition of the real numbers. The real numbers are defined as a Dedekind-complete ordered field. While I understand the components of this definition, i.e the specifications of these axioms, I am absolutely lost as to the philosophical process in place. 
We define the real numbers by a set of properties that we assume to be true.
If someone was to try and construct the set of real numbers using this definition, he would have to turn to the only set that has these properties. Which set is that? It is the set that we assume to have these properties. Well, which set is that? It is the set that we assume to have these properties and so on... I feel like without a concrete reference of some sort all we're left with is cyclical logic.
Can anybody help me shed light on the philosophical process of this definition?

REPLY [15 votes]: It seems like a better question to me than some people seem to think. The question of which set we're talking about doesn't matter, but the question of how we know such a structure exists certainly matters. Lemme paraphrase what it seems to me the issue is, in language that people  here will understand:
Q: I've read that the reals are a complete ordered field. Ok, one can easily show that any two (Dedekind-)complete ordered fields are isomorphic, so this characterizes the theory of the reals, fine. But how do we know that a complete ordered field exists? After all, if there's no such thing as a complete ordered field then the theory of complete ordered fields seems a little pointless.
A perfectly reasonable mathematical question; no need to snicker about the need for a philosopher. Luckily:
A: There are various methods one can use to show that a complete ordered field exists. One of the best known is via "Dedekind cuts", which you can read about in various places online.