Looking for good books for self-study

[sweitzen]

I've been reading up on the history of mathematics and it has motivated me to try to relearn the math I forgot (and more) from a pure mathematics standpoint. My background is in physics. I had the three-semester sequence of calculus for engineers, followed by a semester of linear algebra and ODEs, and a two-semester sequence in mathematical physics. This involved a smattering of topics in analysis and linear algebra, all applied towards the solution of PDEs, some group theory and a little tensor calculus. None of it was proof-based, and all of it was mathematics viewed as a tool, rather than as a discipline in itself, so while I have familiarity with these subjects, I would say I lack any mathematical maturity. (And could stand a solid review of these subjects.)

I have decided to start with set theory, logic and how to read/do mathematical proofs as a jumping-off point.

Two books I ordered to start with are Set Theory and Logic, by Robert R. Stoll and How to Prove It: A Structured Approach, by Daniel J. Velleman.

Can anybody suggest books that may complement those I already have?

For self-study, I like to avail myself of a variety of books on a topic. Some may have excellent exposition, but be a little light in problems. Some may be terse and lack examples, but be wonderfully encyclopedic and make excellent references or resources to drill down deeply into a particular point. Some may be primarily problem books, and some may be introductory or quite advanced.

From the Amazon reviews, Elements of Set Theory by Enderton, Introduction to Set Theory by Hrbacek and Jech, and Basic Set Theory by Vereshchagin and Shen all seem to be introductory to intermediate treatments. Set Theory by Jech seems to be the encyclopedic reference I might want -- not sure how high-level it is. Would any of these titles offer a superior (or, at least, sufficiently different) treatment to Stoll to warrant its addition to my library?

Opinions and other suggestions welcome.

 

[Stephen Tashi]

So many books! f you want to follow a program that duplicates what math majors study, don't go overboard on set theory and logic. Of course, there are specialists who spend their career in set theory and logic, but they are a minority. You have to know enough set theory and logic to avoid mistakes in reasoning and to recognize common approaches to proof ( -for example, proving set A = set B by proving A is a subset of B and B is a subset of A). However, you won't make much progress in math if you devote yourself to getting an absolutely perfect low level foundation.

 

[micromass]

The books you mention are for a deep knowledge of set theory. You won't need all of it to be a decent mathematician. In fact, I highly recommend doing advanced set theory only after you've seen other areas of mathematics!

What I recommend is to do the first chapter of the topology book of Munkres. This has all the set theory that you'll need for the first few math courses...