``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.''
I have put here the current version of the notes from which I am lecturing the first part of the course, on constructive logic; I have put here the current version of the notes from which I am lecturing the second part of the course, on model theory. Other parts will appear later.
I have linked from my
Materials for Part II Mathematics page a lot of nice stuff about
computable functions, lambda calculus etc. [Scroll down to ``Languages
and Automata'']. Some of it is a bit basic for Advanced Lifeforms such
as your good selves, and of course the links are on that Part II page
beco's they are aimed primarily at Part II students, but there is some
material there (mainly the lambda calculus) which is not lectured at
Part II, and in any case a bit of revision will do you no harm.
Speaking of lambda-calculus, here is Toby Miller's nice lambda term that tests equality of Church numerals. And I found this rather nice lambda-calculus reduction workbench. I hope you will find it fun.
Here are Frank Stephan's Notes on computation theory whence I lifted the proof of Kleene-Post.
Another book possibly worth getting is Peter Smith's Gödel book .
In 2016 we had a couple of guest lectures from Maurice Chiodo (who is here for a couple of years on a Marie Curie postdoc) on Automatic groups. Readers might like to look at Automatic Sequences, theory, applications, generalizations by Allouche and Shallit. I shall be pillaging material from it, but I don't know how much! You might like to be ahead of the game, and have a look at it. It certainly looks like fun.
In 2017/8 as in 2016/7 I plan to cover some constructive arithmetic and possibly some constructive Analysis (much harder!), with the intention that my students will look at familar Part I material In A New Light. I have just (re)discovered the entirely delightful A Primer of Infinitesimal Analysis by John Bell. Worth a look, tho' perhaps not worth buying.
Wilfrid Hodges: Model Theory (either the five minute argument or the full half-hour) is something the Logician-About-Town should have on their shelves.
All the above books are Cambridge University Press and so you can get 15% off the list price at the Cambridge University Press bookshop in Trinity street [allegedly the oldest bookshop in the world] by brandishing your wee blue card.
Also by John Bell (co-authored with Alan Slomson) but not published by CUP (it's from Dover) is Models and Ultraproducts. It was the first book on model theory that I read, and I loved it. And it's just the right level for you lot, too.
Large Countable Ordinals;
BQOs and WQOs;
Quine's Set Theory ``NF''.
I welcome enquiries about them. That said, the material for the first two might get covered in my Part III lectures if nobody wants to do essays on them, so I might discourage you from doing either of them!
If you are interested in countable ordinals you might like to read (the rather discursive) talk i gave at the TMS in 2012 and Notes on countable ordinals . Cambridge students who did Part II in 2016/7 will have encountered a rudimentary form of Fraenkel-Mostowski models in my proof of the independence of the Axiom of Choice. I promote Fraenkel-Mostowski models partly because the final step in Randall Holmes' proof of the consistency of Quine's New Foundations is a [diabolical] FM construction and it is a medium-term project of mine to really understand it. I am most of the way there, but the one part I have not yet fully mastered is precisely that final step. So I am always happy to talk to anyone about FM constructions! That diabolical FM construction is of course beyond the scope of a Part III essay [and i use the word diabolical advisedly: Holmes had to sell his soul to The Auld Ane to get the proof, and we owe it to him to ensure that his sacrifice is not in vain] but the general idea of an FM construction is a very good topic for a Part III essay. If we have several takers we could have a reading group and that could be Serious Fun.