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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.

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  • $\begingroup$ The notion of a single domain of discussion in a formal mathematical statement is usually too restrictive. Every quantifier in a statement could be restricted to a different, possibly empty set, e.g. $\forall x\in R: \forall y \in N:$... In logic, the rule of Universal Specification can be used to conveniently introduce of new free variables. This only works if you assume a non-empty domain of discussion. ... $\endgroup$ – Dan Christensen Oct 2 '15 at 20:22
  • $\begingroup$ Since quantifiers in mathematics can be restricted to possibly empty sets, Universal Specification is not generally used in formal mathematics to introduce new free variables. So, there is no need for domains of discussion in mathematics. $\endgroup$ – Dan Christensen Oct 2 '15 at 20:25
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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.

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  • $\begingroup$ That book looks very interesting! And yes, I have to review set theory but it was no fun :( So I was rather thinking of learning new stuff and reviewing as I study. If you have any recommendation on a book focused on logics, that would be helpful, too. $\endgroup$ – Taxxi Oct 2 '15 at 11:24
  • $\begingroup$ I wish I had something like that. But most, if not all, of what I had learned about logic was learned "on the job" (be it teaching, or while working out specific problems)... $\endgroup$ – Asaf Karagila Oct 2 '15 at 11:27
  • $\begingroup$ Hm, that approach doesn't sound bad. $\endgroup$ – Taxxi Oct 2 '15 at 11:35
  • $\begingroup$ Sure, it's just not for everyone... $\endgroup$ – Asaf Karagila Oct 2 '15 at 11:39
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Logic textbook "less verbose" and quite general :

As an alternative (with less topics covered) :

Set theory :

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  • $\begingroup$ I've briefly checked the contents. Books by Derek Goldrei seem to be nice but I think I've already studied most (probably all) of them. Reviewing the previous results is secondary for me since I can do it as I learn new results. The first books seem to have additional topics I haven't learned before, which is nice. $\endgroup$ – Taxxi Oct 2 '15 at 7:16
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You could have a go with H-D Ebbinghaus, J Flum: Finite Model Theory and T Jech: Set Theory.

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  • $\begingroup$ Thanks for the recommendations. I want to ask if there is any reason I should study 'finite' model theory, because I haven't learned model theory itself in depth before. $\endgroup$ – Taxxi Oct 2 '15 at 7:33
  • $\begingroup$ One possible motivation could be that it has applications to databases. For model theory in general, you could also look at the introductory book by Prestel and Delzell Mathematical Logic and Model Theory . $\endgroup$ – Martin Peters Oct 2 '15 at 8:25
  • $\begingroup$ Fair enough. Probably I'll look into both of them. $\endgroup$ – Taxxi Oct 2 '15 at 8:43
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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!)

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

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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

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