# What kind of set-theory is sufficient to understand mathematical analysis?(book recommendation))

I am looking for books with set theory and logic that is sufficient to understand mathematical analysis. I guess another question might be if there even exists such a book.

There are basically two problems I have seen in real analysis that requires set theory. They often create very big sets, but in set theory you can't just create sets, you have to know why it is a set, in order to not get a paradox? The second thing from set theory that is often used is the axiom of choice and zorn's lemma.

Are there more things from set theory that is used in real analysis?(and also functional analysis)(apart from the operations of unions, intersections etc..)

Are there any books that gives a good(and hopefully easy) introduction to all that is needed of set-theory in mathematical analysis?

• Maybe Set Theory and Metric Spaces by Kaplansky. My Analysis professor recommended it to me when I asked him a similar question, and I found it very useful. It is not a giant book of set theory, rather it is about the set theory that is required for analysis. – MY USER NAME IS A LIE May 21 '15 at 18:18
• For a quick introduction to formal logic and axiomatic set theory (the parts you may actually use in mathematical analysis), you may find useful the tutorial that comes with my proof checker available at dcproof.com – Dan Christensen May 21 '15 at 19:58
• There is probably a lot of set theory you won't need. Don't worry too much about obtaining a contradiction. You can safely construct new sets from old sets by any of the following operations: (1) Selecting a subset of another set. Just don't refer to the new set in your selection criteria. (2) Obtaining the Cartesian Product of 2 or more sets. (3) Obtaining the power set of a set. (4) The pairwise union of two sets. (5) The union of a family of sets. (6) The axiom of choice to obtain a choice function. The most complicated. See: en.wikipedia.org/wiki/Axiom_of_choice#Statement – Dan Christensen May 21 '15 at 20:05

Try the first chapter of Topology by Munkres.

• Thank you, I have just one question, do you know if that text contains info about why some sets are legal and why some sets are not, and why the sets created in real analysis(or topology) are legal? – user119615 May 21 '15 at 18:22

Two very standard texts on set theory are

Introduction to Set Theory by Hrbacek & Jech.

This book approaches the subject informally(not much formal logic) and has a good range of topics.

Also there is

Elements of Set Theory by Enderton.

This book requires some familiarity with formal logic and so it a bit more rigorous than Hrbacek & Jech. It doesn't cover quite as many topics as the first book, but does cover anything you would need for real analysis.