``Good afternoon ladies and gentlemen. The subject of mathematical logic splits fourfold into: recursive functions, the heart of the subject; proof theory, which includes the best theorem in the subject; sets and classes, whose romantic appeal far outweigh their mathematical substance; and model theory, whose value is its applicability to, and roots in, algebra.''
If you are interested in pursuing set theory at all then you may well be curious about the foundational role of set theory and wonder where the axioms came from and what they are supposed to do. An Introduction to the Axioms of Set Theory (which is the current draught of a short book I am writing for CUP) just might be part of what you are looking for.
You might be interested in writing the essay on Set Theory Without the Axiom of Foundation or perhaps the essay on Better Quasi-orders .
This year (2006/7) the course is entitled `Logic and Set Theory'. The appropriate extract from the Part III handbook is here . The prerequisites for this are all provided in the Part II ``Logic, and Set theory'' course - which I used to lecture and which is, by a happy coincidence, covered in my book Logic, Induction and Sets . Professor Johnstone's book "Notes on set theory" is good too. Raymond Liu has kindly allowed me to post here his notes of Professor Hyland's Part II Set Theory and Logic lectures . It is Professor Hyland who is currently lecturing this course.
The notes for my lectures come in two main files. The bulk of it - everything except BQO theory and the theory of computable functions - is here . These have been weeded of not only the messages to myself, and false starts etc etc but even the assorted fun things that aren't examinable but which it breaks my heart to leave out. The sacrifices one makes...
The notes on BQO theory have not been as thoroughly weeded.
I shall also cover some of the Theory of Computable functions but I have no plans to put here any materials relevant to it. This is adequately covered in Logic, Induction and Sets and Notes on Set Theory. I can't think of anything I can usefully add to chapter 6 of Logic, Induction and Sets.
Here is the pdf file of the independence of the axiom of extensionality .
I found this rather nice lambda-calculus reduction workbench. I hope you will find it fun.
The choice-free version of the Erd\"os-Rado theorem is not in the notes as I'm writing it up for publication, and here is the version I have sent off to the JSL.
Here are my notes for the CS minicourse on Countable ordinals. I will undoubtedly lecture at least some of this stuff.
I will probably not get round to lecturing in detail the logical roots of Ramsey's theorem. Even if I do, punters will probably appreciate having it set out in idiot-proof form. This link: A commentary on Ramsey's original paper is to my notes-for-myself on those roots. (I am the idiot in question, and these notes are messages to myself designed to supply what is needed for refreshing my memory every time i forget it).
This year for the first time i intend to cover the theorem of Frayne, Morel and Scott that elementarily equivalent structures have isomorphic ultralimits. Here is the handout.
I hope Timothy Chow won't mind me putting in this link to his notes of forcing for beginners.... (If this link doesn't work for you, edit the window to remove the reference to dpmms. Not sure why my server puts that in ....)
Wilfried Buchholz has allowed me to use his very cute proof of the multiset ordering theorem. Read it and improve your soul. The proof that Rieger-Bernays permutation models preserve stratified formulae is in, for example my Church festschrift paper . The proof that they preserve the axiom of replacement i am leaving as an exercise.
Here is an explanation of why the sets hereditarily of finite support form a model of ZF (minus foundation and choice)
For the moment there appears to be only one item not directly connected with this year's Part III course:
My commentary on Henkin's papers on omega-incompleteness