Foundations of set theory

Abstract

Abstract The topic of our conference is truth in mathematics, and certainly the implied question is in part: what constitutes truth in mathematics where, in contrast to the natural sciences, there are no phenomena to be saved? In the fourth century BC, Plato answered this question by observing that we begin with the idea of a certain structure, perhaps derived from experience, and, by analysis (which he called ‘dialectic’), we arrive at the principles which define this structure. What is true of the structure then is what can be derived from those first principles. This prescription works very well in the cases of Euclidean geometry, the theory of real numbers, and the theory of finite sets (or of sets of rank less than the least inaccessible cardinal, etc.), where we have ultimately agreed on certain (second-order) axioms which characterize the structures in question to within isomorphism. However, even in these cases, the status of certain propositions about these structures — for example, the Continuum Hypothesis — remains undetermined, pointing to an incompleteness of the laws of the underlying higher-order logic and, in particular, the laws governing the logical notion of set (to use the term of (Shapiro 1991)) implicit in the second-order comprehension axiom. In another direction, Godel’s incompleteness theorems yield propositions in predicate logic of any order which are not provable in that system, but which are provable l?y passing to logic of still higher-order. In this direction we are led to the theory of transfinite ordinals and to set theory; in particular, we are led to the question of the existence of large cardinals. However, in this case, as opposed to the case of Euclidean geometry et al., there does not seem to be just one generally accepted idea of the universe of set theory and of the laws defining it. Thus, perhaps there is no one notion of truth in this subject; there may be different conceptions, each carrying its own notion of truth-as in the case of different geometries.