Formal Semantics and the Existence of Sets
Michael Jubien · Noûs · 1981
Nearly everyone who has any belief at all in the matter thinks that if one adopts a interpretation for a formal theory, one thereby becomes committed to the existence of (certain) sets. The main purpose of this paper is to argue that this belief is false. It seems to me that its falsity follows from a few surprisingly straightforward points about the semantics of formal theories. If this is correct, it may be of methodological interest to those who are suspicious about the existence of sets. They may needlessly have deprived themselves of the use of set-theoretical semantics. A second goal of this paper is to show how a certain problem about pure set theory can be solved by applying the considerations raised in the first part of the paper. I call this the problem of a realistic interpretation of set theory. This is more likely to be of interest to those who have no doubts about the existence of sets. The first part of the paper presupposes only a basic familiarity with (countable) first-order languages and theories, and with (some) one of the usual set-theoretical (or: model-theoretical) notions of interpretation for these systems. In particular, I treat interpretations as ordered sets whose first members are non-empty sets, called of interpretation, and whose other members are correlated with whatever non-logical symbols may be present in the language (and are themselves closely related to the domain of interpretation). Such ordered sets are perhaps most elegantly viewed as functions whose domains are small countable ordinals, where functions are in turn viewed as sets of ordered pairs, and ordered pairs are regarded systematically as certain unordered sets. But such details are unimportant beyond the central fact that in the end an interpretation of a first-order language simply is a more or less complicated set one of whose ingredients is a domain of interpretation for the quantifiers of the language. The second part of the paper presupposes a rudimentary acquaintance with Zermelo-Fraenkel set theory (ZF). I assume in both parts of the paper that ZF is consistent.