# Book recommendations for studying mathematical areas based on set theory

I am at the end of my studies with set theory, and I would like to continue in fundamental fashion, and study for example calculus based on set theory. So, I am talking about not calculus the way it is studied in college, but calculus studied from the set theory perspective. Every time a theorem is talked about or else, it should be said how it relates to set theory.

I am looking book of that type, so please let me know if you know some.

Thank you

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This isn't related to what you're looking for, but I feel like it might interest you: non-standard analysis. – Git Gud Jun 2 '13 at 9:14
I'll assume that with "at the end of my studies with set theory" you mean "finished a course in set theory" as opposed to "a master of all that is in Jech's book". At this point you might find model theory, mathematical logic, universal algebra, or even category theory appealing. – András Salamon Jun 2 '13 at 9:20
How about learning more set theory instead? – Asaf Karagila Jun 2 '13 at 9:23
I don't think a text of this nature exists. At best (or worst, depending on your viewpoint) one would construct the real line (or something similar) from the axioms of set theory, and then prove that this construction has certain nice properties, and then continue to prove results using already-proved facts, but generally not from the axioms of set theory. Saying that, the set-theoretic study of the real line is a quite important topic in (modern) set theory, but here the focus is not on calculus, per se, but rather on the combinatorial nature of the reals (and associated concepts). – arjafi Jun 2 '13 at 9:23
If you view set theory only as a foundation of mathematics, then it is either assembly language or machine code. However set theory as a mathematical field is just like any other. It is filled with its own conventions, and practitioners take short-cuts with notations, use intuition and pictures to convince themselves of "truth", and write their proofs using words and not only symbols. The only difference is that set-theorists are usually after the epistemological limits of their field in a manner that most (but certaintly not all) other mathematicians are not. – arjafi Jun 2 '13 at 9:42

It seems to me that you are trying to understand the set theoretical foundation of calculus. This would be equivalent to learning how to program in C++ and then insisting to learn how the CPU interprets the compiled code, and how the compiler works.

It is a useful knowledge, but not very useful for C++, or in this case -- for calculus.

If you wish to learn more about the interactions of set theory with other fields of mathematics, I suggest that first you get comfortable with following set theoretical related topics:

1. Descriptive set theory,
2. Basic topology,
3. Cardinal arithmetic and basic PCF theory,
4. Model theory.

Then you can apply these into measure theory, which is the modern extension of calculus; set theoretical topology; abstract algebra (many courses in advance model theory basically amount to algebra and algebraic geometry).

Studying these topics could take a couple of years, and by then you may find yourself interested in set theory per se. Let me give some basic recommendations for books.

1. Moschovakis - Descriptive Set Theory.
2. Engelking - General Topology.
3. Holtz, Steffens, Weitz - Introduction to Cardinal Arithmetic.
4. Chang, Keisler - Model Theory.
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Basic topology?! I think you mean general topology. There's nothing basic about it! – arjafi Jun 2 '13 at 9:46
@Arthur: Of course I meant general topology, but I specifically meant the basic parts - roughly what they cover in introduction courses to topology, up to (and not including!) basic algebraic topology. At least as a start... – Asaf Karagila Jun 2 '13 at 9:48
In Classical Descriptive Set Theory there is also a book by Alexander S. Kechris, but I don’t know how is this book comparable to others. – Alex Ravsky Jun 2 '13 at 10:06
@Alex: Kechris' book is also pretty nice, but I prefer Moschovakis. At least to begin with. Kechris' book is more topology oriented, in some sense. Moschovakis was a nicer read. – Asaf Karagila Jun 2 '13 at 10:07
@AsafKaragila: Thanks for the explanation. I am more topology oriented too. :-) – Alex Ravsky Jun 2 '13 at 10:09

There is a classical series of books by Nicolas Bourbaki, but, personally I dislike “his” exposition as somewhat cumbersome, and, therefore, not very clear.

Also I remember two books based on set theory: “General topology” by Ryszard Engelking (I completely read this book and I call it a Bible of a general topologist :-) ) and “Algebra: rings, modules and categories” by Carl Faith (I don’t read this book).

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