Recommendations for Intermediate Level Logics/Set Theory Books

Question

I currently don't know what books I should read after having studied elementary logics/set theory, so I'd be glad if I can get some recommendations. My classes used the following two books for logics/set theory respectively:

• A Mathematical Introduction to Logic, Herbert B.Enderton
• Introduction to Set Theory, Karel Hrbacek/Thomas Jech

We've dealt up to choice of axiom/ordinals in set theory, and soundness, completeness in first-order language/computability/proof of Gödel's incompleteness theorem(+a bit of model theory) in logics class.

The problems I have now:

1. I studied set theory first, and had a hard time understanding why every proof seems to be more about logics behind it. Some of the doubts became clear after I've studied logics, but still the connection between logics and set theory is rather unclear to me. For example, what is the universe(as in a first-order language) of set theory?
2. The logics textbook was way too verbose, and sometimes introducing a concept only naively, thereby brining more confusion. On the other hand, the set theory book was a bit tough for me as a beginner.

So basically, my brain is full of all the results about logics/set theory but not in a coherent way. I tried to read other introductory level books but it became too tiresome for me. As my experience says, I think it's better to grab a book which deals logics in a more general settings to clear my doubts; probably not just limiting the topics to first-order language. After that, reading an intermediate level set theory book would be nice, I thought.

I'd much prefer dry books especially in these fields rather than verbose ones. Any recommendations are welcomed.

Answer 1

I like Lorenz Halbeisen's "Combinatorial Set Theory" book. It also gives some basic introduction to logic, and how it is used in set theory. The book itself is very thorough and the parts I have read were mostly well-written.

Let me also add, that if you felt a bit shaky on the way logic was used in the set theory proofs, then perhaps it's best not to skip proofs that you already saw. Instead read them more thoroughly to find out new gems of understanding.

Answer 2

Logic textbook "less verbose" and quite general :

• Dirk van Dalen, Logic and Structure (5th ed - 2013).

As an alternative (with less topics covered) :

• Derek Goldrei, Propositional and Predicate Calculus : A Model of Argument (2005)

Set theory :

• Derek Goldrei, Classic set theory (1996).

Answer 3

You could have a go with H-D Ebbinghaus, J Flum: Finite Model Theory and T Jech: Set Theory.

Answer 4

I present the main picture of the foundations of mathematics in very clear, rigorous and concise ways in settheory.net

Answer 5

There is a widely used extended Study Guide to intermediate/advanced textbooks in logic available here

In particular, §4.3 of the Guide is on introductory set theory books; the whole of Chapter 7 on more advanced texts.

You should be able to find something suitable to your interests/level there. (As Asaf says, Halbeisen's Combinatorial Set Theory is good and well put together: if/when I get round to updating the Guide, it might well get "promoted" higher up the list of recommendations!)

Answer 6

Judith Roitman's "Intro to Set Theory" is excellent. While it begins at the beginning, the pace is fast with filters and ultrafilters quite early on. Also model theory is integrated along the way, so not long into it, one encounters $V_{\omega}$ modeling ZF.

At the end there is a section on "semi-advanced " ST with partition calculus, trees, CH and MA.

Beautifully written with many cheerful facts intersperced along the way, even for what might otherwise be considered more beginning material.

Generously available for free: https://www.math.ku.edu/~roitman/SetTheory.pdf