Set Theory and Nominalization, Part I

This paper argues that the basic problems of nominalisation are those of set theory. We shall therefore overview the problems of set theory, the various solutions and assess the innuence on nominalisation. We shall then discuss Aczel's Frege structures and compare them with Scott domains. Moreover, we shall set the ground for the second part which demonstrates that Frege structures are a suitable framework for dealing with nominalisation. 1 The Problems We shall examine the problem of the semantics of nominalised terms from two angles: the formal theory and the existence of models. 1.1 The problem of the formal theory Any theory of nominalisation should be accompanied by some ontological views on concepts | for predicates and open well-formed formulae act semantically as concepts. This is vague, however, if only because where I use the word concept, someone else might use class, predicate, set, property or even system(Dedekind). This terminological profusion is hardly surprising, for we are touching on the problem of universals, a problem philosophers have been debating for hundreds of years. (This new term | universal | may be more confusing than any of the others, but we may use Aristotle as a preliminary guide and deene a universal to be that which can be predicated of things.) The aim of this section is not to take a standpoint on any of the philosophical theories of universals; rather it is to show that, no matter what approach we adopt, nominalisation is going to generate a problem.