(While we are on this subject I should mention that Wilfried Buchholz has allowed me to display here his very cute proof of the multiset ordering theorem. Read it and improve your soul. )
The current thought is that if there is to be a demand for a reading course in set theory then the book-to-be-read will be Thomas Jech, Set Theory available from Springer.
Last Year's (Set Theory) Lecture Notes are are provided, at readers' own risk. 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.
The choice-free version of the Erdös-Rado theorem is not in the notes as I'm writing it up for publication, and here is the version that has now appeared in the JSL.
Here is my Tutorial on Countable Ordinals .
I hope Timothy Chow won't mind me putting in this link to his notes of forcing for beginners.....
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