Mathematical Existence in the Theory of Sets

Abstract

The purpose of this paper is to trace some of the developments arising directly out of the crisis in set theory, in terms of the various views they embody of the nature and existence of their subject matter. Three viewpoints particularly emphasized were the platonistic, the phenomenonalisic, and the intuitionistic. Nominalism, a philosophical position often mentioned in connection with mathematical existence, has been left out of the discussion, primarily due to the fact that there has - as yet - been no set theory constructed along purely nominalistic lines. The philosophical perspectives are introduced in section I. Much of the material presented in this section was drawn from essays in Newman's The World of Mathematics; it represents basic ideas rather than the complete personal philosophies of the individuals cited. In Particular, it is not claimed that the intricate metaphysics or A. N . Whitehead is at all adequately represented by his argument on the reality of the abstract. The other main source used in this section is Bergson's "An Introduction to Metaphysics," While not in the mainstream of mathematical thought, even of the intuitionistic school it was chosen to represent, this work provides valuable insights into that world view which denies the significance of expressible concepts and finished objects or thought. It is at worst an interesting background to the anti-formalism or institutionalistic mathematics. Section II describes briefly the crisis or nineteenth century mathematics, and Section III sets forth explicitly some or the paradoxes which undermined naive set theory. Sections III and IV present the most important of the formal systems which have been designed to avoid these contradictions.