# Faculty of Mathematics

This area of courses is more unconnected than others, so please look at each course separately:

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## Category Theory

The Category Theory course is not really a Logic course, though it sits better with the logic courses than with any others. Category Theory is very useful for nearly all branches of pure mathematics. The only prerequisites for this course are a sufficient knowledge of some kind of pure mathematics, and some familiarity with abstract algebra. A first course in Logic might be helpful.

### Useful books and resources

If you are happy with your level for all the other courses you plan to do, you may want to have a little preview of what Category Theory is about. Dr Goedecke's teaching page has some lecture notes from the 2013 version of the course. If Prof. Johnstone is lecturing it, the pace will probably be faster. As preview the first chapter might be good to look at, and the first two questions on Example Sheet 1 (2013) given there are fairly easy exercises to get started. See also the videos with some solutions explained to Question 1 lower on the page; but you will only be able to access those when you have your Cambridge IT account.

## Logic

A wealth of detail will be available on the lecturer's course page but that will not be accessible to you until you have a Cambridge IT account.

### Reality check

Part III Logic is conceived as the sequel to the two Part II courses: Logic and Set Theory and Automata and Formal Languages. These two courses have no prerequisites, at least in the sense that the only prerequisite (Numbers and Sets) is in any case compulsory, so if you are a Cambridge student you will have done it. If you are not a Cambridge student you should follow the link and see how it strikes you. Don't worry about the Number theory part - we don't use that in either of the Part III courses described here - but if anything else in that course fills you with alarm then Part III Logic is not for you.

These notes on countability contain everything that (in an ideal world) anyone lecturing Logic and Set Theory at Part II would be able to assume their listeners had learnt in Part IA Numbers and Sets about countability. They contain exercises. We recommend them: you should find them straightforward.

### Prerequisites

If you are still reading this then you are equipped to do the two courses that constitute the suggested background for Part III Logic (namely Part II Logic and Set Theory, and Automata and Formal Languages), and you should consult the course pages for them linked above. You can use their example sheets as a guidance to your study. We recommend that you look at the main questions on example sheets only, excluding any "extra", "additional" or "starred" questions.

Both these Part II courses are lectured in Michaelmas term in 2016/7, and both Part III Logic courses are lectured in Lent term, so - timetabling permitting - there is the possibility of you acquiring the relevant background in Michaelmas.

### Useful books and resources

• The book Notes on Logic and Set Theory by Peter Johnstone (CUP 1987) covers the material of the Part II Logic and Set Theory course.
• If you have a .cam.ac.uk address you will be able to see the potentially useful material on Dr Forster's Part II Materials page.

## Topics in Set Theory

The course Topics in Set Theory presuppose a basic introduction into axiomatic set theory as it is usually covered in the Part II course Logic and Set Theory in Cambridge. The basic material can also be found in Chapter I of Kunen: Set Theory, An Introduction to Independence Proofs. Note that the course will roughly follow Chapters III to VII of that book.

### Reality check

You need to be very comfortable with:

• Zermelo-Fraenkel axioms,
• ordinal and cardinal numbers,
• the basic theory of ordinals and cardinals,
• the Axiom of Choice and equivalents.

You can find a quick self-test with fourteen simple multiple choice questions here. You should easily get most of these right without any effort or preparation. Once you have done them, you can test if you are right by highlighting the text behind each bullet point, which is white so hidden.

• Basics: 1C 2D 3C 4C 5D 6D
• Ordinals: 7B 8C 9A
• Cardinals: 10A 11D 12D 13D 14B

### Prerequisites

In order to gain deeper familiarity with the basics, you can do the relevant set theory examples from Part II example sheets of past years, e.g., the example sheets from the Lent 2015 and the Lent 2016 editions of Logic and Set Theory:

Students who are not happy about the level of their set theory background can consider attending the relevant lectures of the course Logic and Set Theory lectured in Michaelmas 2016. It will begin with ordinals (which you will need) and will close with some Set Theory. If precise timing is a concern for you, check with the lecturer, Dr Forster.

### Useful books and resources

• The book Notes on Logic and Set Theory by Peter Johnstone (CUP 1987) covers the material of the Part II Logic and Set Theory course.