TITLE: Defining $\mathbb{C}$ without defining $\mathbb{R}$
QUESTION [9 upvotes]: To define $\mathbb{R}$, one approach is to start with $\mathbb{N}$ and then systematically introduce $\mathbb{Z},$ $\mathbb{Q}$, and then $\mathbb{R}$. Alternatively, we define $\mathbb{R}$ using its axiomatic definition that it is a complete ordered field. And then, we can construct $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ within $\mathbb{R}$.
Now, $\mathbb{C}$ is introduced in the first way, by making $\mathbb{R}^2$ as a field. Can we follow the latter approach of defining $\mathbb{C}$ axiomatically, and then construct $\mathbb{R}$ inside it?

REPLY [2 votes]: You may find this helpful.
However, I disagree with some of the philosophy embedded in your question. You write:

Alternatively, we define R using its axiomatic definition that it is a complete ordered field. And then, we can construct N, Z and Q within R.

With a small caveat, this isn't really correct, insofar as you can't really "construct things axiomatically" - that's not how math works. In particular, to define a thing, you should either:

Construct it out of things that have already known to exist, or
Characterize it by some constraints and then prove there is a unique thing satisfying those constraints.

Notice the boldface portion above. Suppose you want define $\mathbb{R}$ axiomatically - fine, but you need to prove that there's a unique thing up to isomorphism satisfying your definition. Uniqueness should be easy enough. But how would you prove existence? Well, you'd probably begin by constructing $\mathbb{N}$, and then building your way up to $\mathbb{Q}$ and taking a completion. So this isn't really an alternative. Of course, you could do things differently. For example, you could start of by constructing the surreal numbers recursively, and then obtain $\mathbb{R}$ as a subset of those - that would be a different way of constructing $\mathbb{R}$. But at the end of the day, you still haven't "constructed $\mathbb{R}$ axiomatically" - you've just constructed it in a different way.
The caveat I mentioned earlier is that you can replace the axiom of infinity in ZFC with an axiom that instead posits the existence of a real numbers object, thereby obtaining a new system, let's call it ZFCR. It seems likely that ZFCR is equivalent to ZFC, for the reasons you give.