Materials for Part II Mathematics

I include here a shameless plug for Logic Induction and Sets , the most wonderful introduction to --- well --- Logic, Induction and Sets that the world has ever seen. I'm not suggesting for even one moment that you should read it: just buy it! However, if you really do want to actually read it you will need to refer to the page of typos. It was written to the same syllabus for the old Part II logic course of circa 2000 that also gave rise to Professor Johnstone's Notes on Logic and Set Theory . That course has now bifurcated into the two courses to which this page is devoted: the languages and-automata material from that old course has been hived off into Dr Chiodo's (Now Dr Button's) course where it is joined by some material on PDAs and context-free grammars that was not in the old course (see below); the bulk of the material has gone into ST&L. Old Exam questions going back to the 1990s will still be relevant to people pursuing both courses; such people will benefit from reading either Prof Johnstone's book or mine.

Set Theory and Logic
Professor Leader's lecture notes for this course are available from Gareth Taylor (may he live for ever) and students will find them very helpful. My lecture notes from 2016/7 are here .
These notes on countability (which i drew up for my 1a supervisees) contain everything that (in an ideal world) anyone lecturing Set Theory and Logic at Part II would be able to assume their listeners had learnt in Numbers and Sets about countability. So go ahead, don't be shy.


When lecturing this course in 2016/7 i started by talking about ordinals, so I made available a cleaned up version of my talk on ordinals to the Trinity Mathematical Society from 2012. There is also notes on countable ordinals which might be useful, but they should be read (if at all!) only after the TMS notes.

Additional materials
Here are my supervision notes. The version currently (I am writing in november 2019) visible contains discussions of the questions of Prof. Leader's example sheets from last year. (It also contains a whole swathe of discussion answers to questions on older example sheets going back to Gauss and the Bernoulli family). This year the course is being lectured by Dr Russell, and he may tear up the example sheets from last year and start from scratch, or he may recycle them. When lectures start in the new year i shall delete the current version of this file, and resume my practice of putting into it my discussion answers to the current example sheets, but always with a delay of a couple of weeks; the process of updating the file is clocked by my supervisions: i post my discussions of an example sheet after i have concluded all supervisions on that sheet. Discussions of later sheets (or rather versions of later sheets from earlier years) exist of course and if you are a Part III student from outside Cambridge or have a legitimate need for some for other reasons you can email me to ask for the entire .pdf. (At present it is visible but will be taken down in the new year)

Finally here you can read some discussion answers of old tripos questions. Thery are a bit scrappy, and definitely work-in-progress, so i won't demur if people want enhancements.


Here is Randall Holmes' very nice proof (in ZF) that, for all X, the class of things hereditarily smaller than X is a set (and without using choice!)
Here are some model tripos questions that you might find helpful for your revision, and here is a question more in the style of an example sheet question. These all have model answers which i will show you (or your supervisor) if you submit an answer of your own.
An answer to Hyland sheet 2 q 9 (2007/8 edition). It is a proof that for any vector space all its bases are the same size.
Here is a brief discussion of the subtleties of deducing the axiom of the empty set from the axiom of infinity.
You might like to have a look at my notes (to be published one day, eventually! by CUP) on the axioms of set theory. It lacks the chapter on the axiom of choice, but that chapter is itemised separately here .

Languages and Automata
You will find very useful Prof Pitts' materials on Languages and Automata on the computer laboratory's course pages; ditto his CS 1B materials . There is a slight mismatch between his materials and Dr Button's in that our course covers push-down automata and context-free languages (which CS doesn't - i think!) and we do not cover lambda calculus whereas they do. Nevertheless, Prof Pitts' materials come with the highest possible recommendation. He is sensationally organised and the materials are beautifully set out.
Here are my Regular Languages and Finite Automata materials (originally written for Queen Mary) in pdf format. I am greatly endebted to Chloë Brown for creating a version in html . This version is preferable in various ways to the pdf version, since the answers to the exercises are not immediately visible in the way they are in the .pdf version, but can be seen only when you click on the link. Thank you, Chloë Brown!! Here is a file of answers to the coursework questions at the end .
Some of you have asked me whether Regular languages are good for anything. These notes of Arthur Norman's on the hardness of the equality problem for regular languages may be of interest to strong students.
This discussion answer to Sheet 4 Question 4 Part (i) of Dr Chiodo's 2017 materials might be useful.
Here is a discussion of 2017 paper 4 question 4H.
Here is a discussion of 2017 paper 3 question 11H.
This discussion answer to Sheet 2 Question 1 Part (d) of Dr Chiodo's 2016 materials might be useful.
Here is a discussion of the last part of 2009 paper 6 question 3 of the CS Tripos.
Here is the pdf file of the notes of Richard Crouch's second-year course at the University of Nottingham on Languages, Computation and Automata. They do not correspond exactly to any course here, but students might find them useful: they are very meaty.
I found this rather nice lambda-calculus reduction workbench. I hope you will find it fun.

Materials for 1b Computer Science.
Materials for 1a Computer Science .
Materials for Part III Mathematics .
Materials for Logic-For-Linguists.
Materials for Part II Mathematics .
Materials for Part IV Mathematics .
Materials for the Computer Science M. Phil .