There have been two main philosophical obstacles to investigations in set theory. One is the opinion that there is no conception of the universe of sets that can lead to the solution of outstanding problems such as the continuum problem and the existence of large cardinals. The other is the opinion that set theory is ineradicably 'Platonistic.' Platonism in mathematics, some argue, implies that set theory could be wrong, even though, from an internal point of view, there is nothing wrong with it. They argue that it could be wrong because it implies the existence of things that just do not exist. Both of these opinions, and especially both together, have led to a formalistic attitude toward set theory and, in particular, have encouraged the turn from the study of set theory to the study of models of first order axiomatic set theory. Professor Tait has two aims in this project. First, he wants to argue that the conception of the universe of sets such as is contained in the 1930 paper by Zermelo and further developed in Godel's "What is Cantor's continuum problem" is a coherent conception which supports the axioms of ZF as well as providing a foundation for investigations into the questions of the Continuum Hypothesis and the existence of large cardinals. Second, he wants to examine the philosophical confusions underlying the idea of 'Platonism' in mathematics in light of recent literature on the subject.
