A far too short overview of Set Theory

There is no doubt that sets underpin a lot of the work we do as logicians, mathematicians, and computer scientists, with most of us unafraid to make statements such as “Consider , the set of all formulae satisfying : : : ”, “Take X to be the topological space whose underlying set is : : : ”, or “ The set of possible computations of a Turing Machine : : : .” Thus it is essential for logicians, mathematicians, and computer scientists to know what Set Theory purports to do, how it attempts to do that, what it can’t do, why it can’t do that, and to have a general overview of the development of the subject; after all, we may never know when we might arrive at a problem that is beyond the scope of Set Theory or maybe can be efficiently solved by thinking about the problem set theoretically. These notes are presented as a slight expansion of the very short overview of Set Theory to be presented at the Australian National University’s Summer School in Logic, 1997. The aim is to provide a brief summary of the main areas of Set Theory and to give a flavour of the type of results there and how they may be arrived at. Also, the author hopes that this will act as a warning to those who stray too close to Set Theory so that they can recognise their predicament and consult appropriate references. In no way is this a comprehensive stand alone treatment of set theory. In fact, to keep this document brief we will gloss over a number of non-central, but still important issues; anyone wishing to think more deeply about sets should refer to any one of the texts noted at the end of this paper. Before we get started, a few remarks about our notation. Essentially we will be using the standard notation inherited from first order logic. One type of notation with which you might not be familiar is the use of an overstrike to represent a finite sequence of variables or objects. In particular a = ha0; a1; : : : ; an 1i for some n appropriate to the context. Thus, '(x) is the formula '(x0; : : : ; xn 1) with n variables, x0; : : : ; xn 1 free. Similarly (8v)' is a notation for (8v0) (8v1) (8vn 1)'.