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First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce some advanced topics like the constructible model, the axiom of determinacy and forcing, but not limited to them. The problem is the listeners are philosophy students. They got very little mathematical training. I can not introduce them in a very formal way.

But I still want to impress them by explaining my ideas more intuitively and philosophically. In particular, what's the philosophical meaning behind set-theoretical objects? Are there some theorems, related to mathematical logic, that the philosophy students may have interest?

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Speaking about forcing and determinacy, and other models of set theory issues, without introducing formal definitions is missing the point, unless the point is that the students will have a vague familiarity with the names. There is no "philosophical meaning" behind mathematical ideas, they are not applicable to justice, rhetorics, or demagogy. Mathematical ideas are ideal, perfect, and they are nothing more than ideas. A shared psychosis between all the members of the mathematical society. – Asaf Karagila Dec 19 '12 at 4:07
You might want to pose this question on philosophy stackexchange. I think Asaf has a point about mathematical ideas as inapplicable to justice, rhetoric, and demagogy. However, I think one can maintain that some "philosophical meaning" behind set-theoretical objects lies in their beauty in the same way sculpture or architecture or a good computer has beauty... mostly in terms of order, clarity, and simplicity without over-simpleness. I don't know much about the ideas you've talked about though. – Doug Spoonwood Dec 19 '12 at 4:46
Yes I agree it's difficult to talk about forcing with philosophy students, what for us is beauty and simplicity and clarity may becomes totally mess for them. – Chao Chen Dec 19 '12 at 5:07
I would like to disagree with Asaf and say that set theory has many philosophical meanings, rather than having no philosophical meaning. There is literature on the subject of the philosophy of set theory, by the way, so it might be helpful to google the phrase. – Trevor Wilson Dec 19 '12 at 5:11
@AsafKaragila "Mathematical ideas are ideal, perfect, and they are nothing more than ideas. A shared psychosis between all the members of the mathematical society." Well, there's a philosophical position for you! The kind of thing the OP could discuss in a "Set theory for philosophers" course! Do the techie explorations give any reason for adopting this extreme fictionalism!? – Peter Smith Dec 19 '12 at 9:03
up vote 8 down vote accepted

i) I would check out Michael Potter's wonderful Set theory and Its Philosophy to see the kind of thing that it might be sensible to cover in a course directed to not-very-mathematical philosophers. This brilliant book was written for, inter alia, just such students.

ii) You can introduce the constructible model and talk a little about $V = L$ without losing your audience. But I very, very, much doubt that, if your audience is philosophers with little mathematical background, that there is any point at all in trying to explain forcing. Even Timothy Chow's Beginner's Guide will be beyond them (and coming up with anything more accessible is an unsolved exposition problem). What will matter to your audience, as far as reflection on set theory is concerned, is that certain independence results can be proved, not any of the details of how they are proved.

iii) I would recommend, given your audience, and given they have had a standard intro to ZFC, spending some of your time looking sideways rather than upwards. They need to know (and will be really interested in discovering) that there are other ways of doing set theory: not just Scott-Potter (which arguably fits more nicely the canonical hierarchical picture of the set universe), but say NF.

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Here are a few additional resources that may be of interest:

  • An older but very interesting book, Foundations of set theory by Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Lévy. The book describes the axioms for ZFC but is particularly interested in their motivation and with other philosophical issues in set theory.

  • The paper "Does mathematics need new axioms?", Solomon Feferman, Harvey Friedman, Penelope Maddy and John Steel, Bulletin of Symbolic Logic, 2000. Concerns issues such as whether large cardinal hypothesis should become part of the standard framework for mathematics.

  • A very recent philosphical paper by Joel David Hamkins, "The set-theoretic multiverse", Review of Symbolic Logic, 2012. Hamkins explicitly focuses on the philosophical problem of the objective meaning (or lack thereof) of the phrase "all sets". I think that philosophy students may be able to get something out of this paper if they have been introduced to the continuum hypothesis and L, even if they known nothing of forcing. And it is extremely timely.

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I warmly second, in particular, the recommendation of Fraenkel/Bar-Hillel/Lévy. It is really illuminating about the early days of set theory (how did we end up with the ZFC story we all know and love?). – Peter Smith Dec 19 '12 at 14:09

If your students are still having difficulties with more basic concepts of formal proof, may I humbly suggest my DC Proof program as at least a supplementary resource. The tutorial included with my easy-to-learn proof assistant includes worked examples of, among other proofs, a resolution of the Barber Paradox, its set-theoretic twin, Russell's Paradox, and the related paradox of the universal set -- enough to pique the interest of any philosopher! I also introduce the axioms for the natural numbers and prove by induction that no number can be its own successor. With each of 13 worked examples, I include exercises with hints and full solutions.

Visit my website for more information, a free download and demo video/PowerPoint slides.

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I think it is possible to introduce these ideas to philosophy students without getting formally mathematical.

For example - once such a student understands the importance of axiomatic systems and proof, of foundations, of aesthetic clarity, of the idea of a set - the idea of ZFC comes about naturally without having to introduce the exact formalisation that is used.

Forcing as I understand it (which I don't very much) is a technique in Model Theory. This is a useful distinction to make between Syntax and Semantics (truth interpretation). One can introduce Wittgensteins Truth Tables here from his Tractatus Logico-Philosophicus - a name that they should be familiar with. One recalls (apocraphally) he dismissed mathematics as just syntax - one half of the equation for Model Theory!

One can then introduce Forcing as a technique to create non-standard Models where certain propositions are forced to hold.

Finally, one should introduce them to Philosophers that have or do use these notions. In France - one name stands out - this is Badiou. He not only makes use of ZFC & Forcing but also Sheaves and Category Theory. He is heavy work and he comes from the Continental rather than the Anglo-American Analytic tradition. So this suggestion may not be useful if your st udents are Analytically minded. Having said this, Badiou himself admits that he is trying to create a bridge between these two traditions - so that in itself may be useful.

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