Learning mathematical philosophy

March 5, 2012

The Bristol Formal Methods Seminar is aimed at teaching new postgraduates some of the formal methods used in much of modern philosophy. During the first term we’ve concentrated on logic: thus far the sessions have covered introductory set theory, basic metatheorems of first order logic, and the incompleteness of arithmetic. Next up are modal and temporal logic, while after Easter the focus will shift to formal epistemology and decision theory.

Many of the handouts from these seminars have now been made available on the Munich Center for Mathematical Philosophy’s website, in their section on learning mathematical philosophy. In particular, they have my handout on the compactness and Löwenheim–Skolem theorems.

Since the Formal Methods Seminar isn’t a full-on logic course, I didn’t have time to cover all of the material I would have liked to, and consequently the notes are somewhat incomplete. In particular the downwards Löwenheim–Skolem theorem is not proved, although of course the proof is available in any good logic textbook. I hope to correct this omission at some point, and if anyone has corrections or suggestions, please do .

Update, November 2012

During October, I taught this material again to a new set of Master’s students. Instead of proving the upwards direction of Löwenheim–Skolem I taught the more fundamental downwards direction as a simple application of the completeness theorem, albeit not in its full generality, which would have required more time and technical finesse than I could assume.

The updated notes include this proof and have been restructured somewhat, but otherwise they are largely the same as last year’s.

By Benedict Eastaugh.