B1 Set Theory

This page contains some material that I wrote for the third year course in Set Theory at Oxford some years ago and that may be of interest.

A document about the language of set theory. This handout includes some personal opinions.

A list of the Axioms of Set Theory, in summary form.

For the sake of completeness, the Axioms of Set Theory in the Language of Set Theory.

Once we have constructed the natural numbers with addition and multiplication, we can then go on to construct the integers, the rationals and the reals. Some familiarity with section A ring theory is presupposed. For some reason, this material seems not be on the undergraduate syllabus.

A notational problem raised by Set Theory is how to write aleph. The document contains a digression on the history of the alphabet. Accordingly, here are some links about the Wadi-el-Hol inscriptions, and about writing more generally if you read English or Spanish.

An intuitive discussion of the ordinals, including a brief account of where Cantor got the idea from, and motivation for the operations of ordinal addition and multiplication.

A discussion of the Axiom of Replacement, including a brief account of why it's controversial.

A little bit more about ordinal arithmetic.

A short list of what various axioms are needed for.

Further reading

Rudy Rucker's popular-science book Infinity and the Mind is (among other things) an accessible introduction to set theory.

An interestingly different way of establishing Set Theory, together with some discussion of the history and philosophy of the subject, can be found in Potter's book Set Theory and its Philosophy.

For some background, you may find JW Dauben's biography of Cantor interesting.

Some of you will already have read Frege's Foundations of Arithmetic. If you haven't, I highly recommend it. It used to be universally held that Frege's approach to the Natural Numbers was holed below the waterline by Russell's Paradox. Some people now think that Frege's argument can be fixed. Certainly, to me, Frege's text looks prophetic.

For a recentish survey of the area (at research level), see The Philosophy of Mathematics Today, edited by Schirn.

Frege's book was, of course, written in German, as Die Grundlagen der Arithmetik.

More about the arithmetic of transfinite numbers is contained in Sierpinski's book Cardinal and Ordinal Numbers. There is one copy of this in the RSL stack, and if you are feeling very rich it is possible to get it on Amazon.

A more elementary, and informal, introduction to Set Theory is available in the book in the Schaum Outline Series.

What comes next?

If you're interested in studying further, you might enjoy the section C Axiomatic Set Theory course. It consists of Gödel's proof that if the ZF axioms of set theory are consistent, then they are consistent with the Axiom of Choice and the Continuum Hypothesis. You will need to be fairly comfortable with the material in B1 Logic.

Beyond that, there's Cohen's proof that if ZF is consistent, then so are the negations of the Axiom of Choice and the Continuum Hypothesis. This does not fall within the scope of the undergraduate syllabus (sadly); there are various books that cover this, including books by Kunen, Bell, Jech, and Smullyan and Fitting.

This page last modified by R. W. Knight
2nd October 2018
Email corrections and comments to knight@maths.ox.ac.uk

Last modified: Wed Dec 3 11:07:30 GMT 2008