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
 A: 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:


*

*Descriptive set theory,

*Basic topology,

*Cardinal arithmetic and basic PCF theory,

*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.


*

*Moschovakis - Descriptive Set Theory.

*Engelking - General Topology.

*Holtz, Steffens, Weitz - Introduction to Cardinal Arithmetic.

*Chang, Keisler - Model Theory.

A: 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).
