On the construction of the number systems Recently, I read something about how set theory is used to built the number systems (starting with $\mathbb{N}$ and building the sets $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}.$)
I would like to know more about this matter and for that I have two questions.

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*What is the necessary mathematical background to start studying this subject? And what books do you recommend as preliminary?


*After having some good notions about the preliminary, can you point me out a really good book about the construction of numbers systems?
In short, what path should I follow for a good understanding about this topic?
Any suggestion is welcome!
Thank you for your attention!
 A: The only prerequisite is the proverbial "mathematical maturity". To be a bit more specific, you should be comfortable with the basics of naive set theory. Just to cite a few examples:

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*Relations as sets of ordered pairs.

*Proof by induction.

*The connection between equivalence relations and partitions.

*How a surjective function $A\rightarrow B$ is "essentially the same" as an equivalence relation on $A$.

That's not meant to be a comprehensive list.
There's a famous book that covers the construction of $\mathbb{R}$ from $\mathbb{N}$: Landau's Foundations of Analysis. It's written in a severe Definition-Theorem-Proof style that people used to admire. Personally, I don't recommend it for beginners.
For a more user-friendly source, I'd recommend Enderton's Elements of Set Theory. This covers the preliminary material you need, before constructing the natural numbers in Chapter 4 and the integers, rationals, and reals in Chapter 5. (Probably you can find a free pdf of the whole book online, but with a quick look I've found only Chapter 5.)
I should also mention Dedekind's original paper, "Stetigkeit und irrationale Zahlen" ("“Continuity and Irrational Numbers"), giving his construction of $\mathbb{R}$ from $\mathbb{Q}$. Dedekind is a remarkably clear writer. Of course, his notation is outdated in places, though when I read it (many years ago), I don't recall that causing any hiccups. However, looking for an online copy (available at Project Gutenberg), I ran across this paper: An Examination of Richard Dedekind's "Continuity and Irrational Numbers", Rose-Hulman Undergraduate Mathematics Journal, by Chase Crosby.
Yet another source from a very good writer: the epilog to Spivak's Calculus. He gives three constructions of the real numbers, namely Dedekind cuts, Cantor's fundamental sequences, and infinite decimals. The second two are presented as exercises with detailed hints. He also shows the essential uniqueness of the real numbers.
Three more remarks: (1) Category theory sheds new light on some of this, specifically what "essentially the same" means. (2) The construction of $\mathbb{N}$ as von Neumann ordinals can seem rather artificial. What really matters is not what natural numbers are, but the axioms they obey. (3) This is true also for the later stages of the construction. Specifically, there are two famous constructions of $\mathbb{R}$ from $\mathbb{Q}$: Dedekind cuts, and Cantor's fundamental sequences. They look pretty different, but you end up with isomorphic structures.
