Last fall I taught a course in the Philosophy of Set Theory at NYU and you can find the reading list available on my web page. This course was more narrowly focused on the question of realism and pluralism than some of the other syllabi mentioned in the other excellent answers here.
Update (12 years later). Let me recommend my book:
- Hamkins, Joel David, Lectures on the philosophy of mathematics, MIT Press (2020). ZBL1511.00002.
I wrote this book in connections with my lectures on the Philosophy of Mathematics at the University of Oxford, 2018-2022. There is also a video series available on YouTube, Oxford Philosophy of Mathematics. I have used this book to teach numerous times.
My approach to the subject is grounded in mathematics and motivated by mathematical inquiry and practice. The chapters are organized around mathematical themes, such as Numbers, Rigour, Infinity, Proof, Computability, and so forth. The last chapter is on Set Theory, which covers various foundational issues, including all what one expects regarding Frege, Russell, Zermelo, axiom of choice, continuum hypothesis, cumulative hierarchy, and so forth, but also getting into the debate on pluralism as well as the significance of the large cardinal hierarchy, including intrinsic versus extrinsic justification and the criterion by which we should adopt new axioms, the debate on strong versus weak foundations, and more.
