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I am having a lot of problems in understanding my professor's notes on Set Theory. The main problem is: they are basically a collection of proven results, and do not provide context/ motivation/intuition about what is going on. Some proofs are also elliptic, but it is not quite the problem. The fact is I am getting to hate this subject that should be so profound and foundational. Mainly, it seems to me a collection of very technical results.
Our course includes: axiomatics of set theory (ZF and MK mainly), ordinals and cardinals, recursive constructions and the Von Neumann hierarchy, cardinal arithmetic (also in a choiceless environment) cardinal exponentiation, cofinality, regular and singular cardinals, stationary and club sets.
I am quite ok up to Von Neumann hierarchy, then I really get lost. I repeat, technical results and profound ones really get mixed, and the impression is to be learning a list of unintuitive facts rather than understading: I mean, the reasoning appers highly destructured. This is the main fault.

If someone could suggest me some coincise notes or textbook I would be eternally thankfull. I am not criticizing no one, but I am a beginner and I am not getting any meaning here.

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    $\begingroup$ Sadly, no. Almost all the topics I cited are surely not naive set theory, as they are strongly based on axiomatics (think the importance of the axiom of foundation in many facts about ordinlas). Many are also non elementary I think. $\endgroup$
    – Francesco Bilotta
    Dec 13, 2019 at 17:20
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    $\begingroup$ In my (probably not universally shared) opinion, it is unwise to expect anything of set theory that you wouldn’t expect of any other well-developed field of mathematics. It is a technical field and has many facets, each of which may or may not be to one’s taste. I personally liked Kunen’s older book and thought it had good motivation (but it can also be too detailed), but really you should try several if you can. Later, for stuff at the intermediate level that is more “beautiful” than “technical”, I would suggest looking into the basics of large cardinals, particularly ch 17-18 of Jech. $\endgroup$
    – spaceisdarkgreen
    Dec 13, 2019 at 17:25
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    $\begingroup$ But in case my first couple sentences seem too negative, I should add the obvious point that like any subject, one shouldn’t necessarily give up just because of one instructor’s presentation. So it’s good to look for supplements, as you’re doing. $\endgroup$
    – spaceisdarkgreen
    Dec 13, 2019 at 17:34
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    $\begingroup$ Have you read this? And since I'm shamelessly self-promoting my stuff, try my lecture notes for a size. $\endgroup$
    – Asaf Karagila ♦
    Dec 13, 2019 at 17:43
  • $\begingroup$ @AsafKaragila thanks I will surely do! $\endgroup$
    – Francesco Bilotta
    Dec 13, 2019 at 17:46

1 Answer 1

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First: I am not a set theorist, but I have a BSc in mathematics, and I am almost done with my MSc (also in mathematics), and set theory happens to be one of those subjects outside of what I do, that I find especially fascinating. I have taken a graduate level course in logic with set theory, but I have only looked at the pure course in set theory and forcing. So, my answer is very much from a students perspective, which I hope is a good thing.

When we did set theory as part of a more advanced course in logic, we mainly used

1.) R. Cori, D. Lascar; Recursion Theory, Gödel's Theorems, Set Theory, Model Theory. Oxford University Press.

This is part II of a series (duology?) of books on logic (first one here). I must say, starting out with axiomatic set theory, I really liked this one, and this is perhaps especially good if you want a somewhat concise yet rigorous introduction.

I also really like

2.) Notes on Set Theory, Second edition, Springer 2006, by Y.N. Moschovakis,

which is of course a more complete book on set theory, but includes axiomatics. Both these books are, in my opinion, concise (at least 1), rigorous, yet accessible.

The main book used in the pure set theory course at our department is

3.) Kenneth Kunen, Set Theory – an Introduction to Independence Proofs, North-Holland 1980

N.B. I have only skimmed this one, however it looks good, and imo, all book recommendations and choices for course literature in logic courses at our department that I've read, has been of very high quality. (We have a very long tradition of logic and many people doing logic related research here, i.a. Per Martin-Löf, so I have great confidence in their suggestions.)

Another book I found useful, that was among the suggested reference literature for the logic course I took was

4.) Thomas Jech, Set Theory, Third edition. Springer 2000

You should be able to read the TOC of each of these books through the links I provided above.

It's always hard to know precisely what someone is after when asking these type of questions, but hopefully this will be of some help.

(Just if you're (or anyone is) interested, but perhaps not that relevant: other suggested, perhaps less well known, reference literature for the set theory course was

Also, perhaps not customary practice, but I must add that: looking through the suggestions by (and by) Asaf Karagila above, these look really good; thank you for those, I will also surely read them.)

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