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.

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.

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