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(At the risk of being vapulated and downvoted, I'll ask this here.)

Suppose you work in a field that has nothing to do with the foundations of mathematics, but thanks to MO, you are becoming more and more interested in topics like axiomatic set theory, the different logical systems (intuitionistic, classical, finitist), category theory, type theory, etc. Thus, you want to understand all of this, and you can spend some time learning from books and articles, without any hurry. But you don't want to actually do research is this field. This is the case for me.

Question: Is there any organized way to achive this? Any book recomendatons to achive this goal? At my university there is no one working in this area, hence I have no one to ask this question directly. My background is naive set theory, naive category theory and some basic logic.

Let me explain with an example where I want to get. Suppose you saw the recent Zizek/Peterson debate, and you have read some of their books, you understood their ideas and you have an opinion. But you can't, and don't want, to sit in front of public and debate with either of them. (Of course, you would like to sit and chat with Zizek for hours :))

Thank you very much.

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    $\begingroup$ This Žižek–Peterson debate? What does that have to do with the foundations of mathematics? $\endgroup$ – LSpice Dec 24 '19 at 14:44
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    $\begingroup$ @LSpice That's why I begun with the phrase "Let me explain with an example where I want to get." You may extrapolate this. $\endgroup$ – EFinat-S Dec 24 '19 at 15:13
  • $\begingroup$ @MattF. I just flipped through that book, and is exactly the kind of book I'm searching for. Thank you very much. $\endgroup$ – EFinat-S Dec 24 '19 at 15:47
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    $\begingroup$ @MattF. : I encourage you to turn your comment into an answer. $\endgroup$ – Timothy Chow Dec 24 '19 at 19:35
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    $\begingroup$ "Of course, you would like to sit and chat with Zizek for hours": no I wouldn't. Notice that for many people he's a charlatan. $\endgroup$ – Quora Feans Dec 25 '19 at 23:36
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One feature of the foundations of mathematics that poses a special challenge (compared to other branches of mathematics) is that it is very easy to get confused about certain distinctions—truth versus provability, theory versus meta-theory, formal versus informal, syntax versus arithmetic, etc. One book that I think is helpful in this regard is Torkel Franzen's Inexhaustibility: A Non-Exhaustive Treatment.

Beyond that, what I would recommend depends a lot on what you're specifically interested in. A topic that comes up quite frequently on MO is reverse mathematics, and for that, I'd recommend John Stillwell's book, Reverse Mathematics: Proofs From the Inside Out. For type theory, I think that Martin-Lof's original writings are still an excellent place to start.

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  • $\begingroup$ <3 torkel, a nice guy i knew online for many years $\endgroup$ – nomen Dec 26 '19 at 21:01
  • $\begingroup$ What is/are the prerequisite(s) for Martin-Lof's writing's on type theory (especially the one you linked)? $\endgroup$ – user 170039 Dec 27 '19 at 4:31
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    $\begingroup$ @user170039: As I recall, there are not a lot of formal prerequisites for reading Martin-Lof's paper, but it will be very helpful to have some prior experience with some kind of sequent calculus for a formal language. Some prior exposure to axiomatic set theory will also help, not so much because he quotes any theorems, but because the motivation for various novel constructions will be clearer if you know how things are traditionally done. $\endgroup$ – Timothy Chow Dec 27 '19 at 15:58
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I recommend Varieties of Constructive Mathematics, a 1987 book by Douglas Bridges and Fred Richman, which covers constructive vs recursive vs intuitionist vs classical approaches, and has a brief introduction to topos theory too, all in 160 pages.

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I think at least one of the following books will be useful to you. I hope your university gives you access to the books available on Springer Link.

Next are two books on logic.

I recommend you read the book "Philosophical and Mathematical Logic", written by Harrie de Swart. It's very good and requires no prerequisites.

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    $\begingroup$ Why is the material on philosophical logic helpful as an introduction to foundations of mathematics? E.g. when we discuss foundations of math here on MathOverflow, we almost never talk about modal logics or the like. $\endgroup$ – Matt F. Dec 24 '19 at 23:48
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    $\begingroup$ @MattF. I think it is important to study philosophical logic because mathematics assumes several axioms of logic. Since there is not only one type of logical system, it is necessary to understand why certain branches of mathematics accept one axiom and reject another (for example, the principle of excluded middle in the case of Constructive Mathematics) . Also, mathematics assumes some rules of inference, so it is worth understanding a little about these things to have a better understanding of the foundation of mathematics. $\endgroup$ – rfloc Dec 25 '19 at 0:06
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One of my students liked Peter Smith's An Introduction to Gödel's Theorem because it takes the time to explain what's going on.

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I was really interested in this stuff as an undergrad, and spent a lot of my free time learning about it. So here is a roadmap, from a well-informed non-researcher.

  1. Enderton's (or Mendelson's) Mathematical Logic. You need the vocabulary, and to start seeing the "standard machines" asap -- e.g., a logic is a language definition, plus semantics, typically expressed using the vocabulary of model theory. In other words, you need to see the interplay between the language and its interpretation.

  2. Get a good advanced model theory book. You're going to need forcing. (But you're also going to need to understand things like the ordinals and transfinite induction, so some solid set theory or even descriptive set theory should come first)

  3. There are a lot of "philosophical" reasons to be interested in constructive logic. They may or may not be interesting to you. If not, at least read up on Brouwer and his intuitionism on a lazy Sunday afternoon. Boolos' Logic, Logic, and Logic has some interesting essays on hairs that shouldn't have been split, but also some good essays on interesting ideas (like the "intuitive" equivalence between a strong set theory and a second order logic).

  4. Definitely, learn a programming language like Haskell or ML. "The Gentle Introduction to Haskell" basically runs through the standard machine to teach you the language. Read that, and implement the "standard prelude" from the types. Ultimately, there isn't a big difference between intuitionism and computationalism, so it helps to understand the limits of computation -- i.e. Godel, Rice, etc.

  5. Category theory is a big topic. Topoi are natural objects that sort of implement the logical notion of a model, in category theoretic terms. If you build up topoi, you can do the "standard machine" in category theory. But there is a lot more to category theory than just this! It's basically "just" a powerful modelling language, so it's used in a lot of different applications.

  6. Don't forget lattices, like Davey and Priestley's Lattice Theory. Denotational semantics are another run through the standard machine, with particular relevance to computational languages.

  7. Depending on the direction you want to go, there's plenty of foundational work you can do in "descriptive set theory" i.e., what kinds of sets can you define measures for, if you do it generating from "simpler" by induction on the ordinals? (This is already the beginnings of a simple stratified type theory, specifically suited for real analysis or game theory, etc.)

  8. The newest "powerful" type theory to have textbooks about it is called "Homotopy Type Theory". Again, another run through the standard machine, this time with ordinal/induction constructions too.

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    $\begingroup$ This is a nice list, but some of these items strike me as unnecessarily advanced for someone who doesn't intend to do research in foundations. For example, I don't think that understanding forcing is really necessary if you're not going to do research. $\endgroup$ – Timothy Chow Dec 27 '19 at 18:20
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Strongly recommend Category Theory for the Sciences (David Spivak) if you are on a budget, you can access the book from the author's page, only difference is that solutions to the problems are not in this version.

I am not a mathematician, but find category theory fascinating, the more I study it, the more I see it as one of the fundamental blocks of mathematics.

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