Seminar Notes

On this page, I'm attaching some links to notes that I have used to teach. I've tried to write them to be useful for people coming from philosophy with a basic background in logic.

In the long term, I'd like to turn them into text books, but for the moment I'd just like them to be useful. They are still in draft form so if you spot typos or errors, I would be very grateful to hear about them. I'm also interested in hearing about possible points of confusion and suggestions to remedy them.

Given that the documents will change, I'll tag the files with version numbers and keep an archive of previous revisions.

Advanced Logic: These notes provide a thorough but swift treatment of Gödel's completeness and incompleteness theorems. Along the way, we also cover some elementary model theory, set theory and recursion theory.

Set Theory: some basics and a glimpse of advanced techniques: These notes are divided into two sections. The first section gives a quick introduction to some of the basic tools of set theory including: the axioms of ZFC; representation of structures and models; transfinite induction and recursion; and Cantor's theorem and CH. The second section provides an overview of some fundamental tools in advanced set theory: constructibility; forcing; and large cardinals.

Truth & Paradox: These notes are about the liar paradox and formal ways of addressing it. We begin with discussion of why the paradox is important and provide a standard formal analysis of it. We then provide detailed descriptions of Tarski and Kripke's solutions to the paradox. [These notes are not yet complete. A more sensitive discussion of coding is required and some proofs at the end of the document need to be filled in.]

A Philosopher's Guide to Forcing (slides): Cohen demonstrated the independence of CH from ZFC by providing a model in which it is false. The construction of the model involves the adjunction of a new, generic object via the technique of forcing. Forcing revolutionised research in set theory but despite this, there remains a certain enigma to the technique: it is difficult to explain why it works. These slides aim to provide a philosophically satisfying explanation of why forcing works and in so doing some analysis of what a generic set is. We simplify the presentation by working with second order arithmetic rather than set theory.