Where do sets come from?

Abstract

Many philosophers take set-theoretic discourse to be about objects of a special sort, namely sets; correlatively, they regard truth in such discourse as quite like truth in discourse about nonmathematical objects. There is a thin “disquotational” way of construing this construal; but that may candy-coat a philosophically substantive semantic theory: the Mathematical-Object theory of the basis for the distribution of truth and falsehood to sentences containing set-theoretic expressions. This theory asserts that truth and falsity for sentences containing set-theoretic expressions are grounded in semantic facts (about the relation between language and the world) of the sort modelled by the usual model-theoretic semantics for an uninterpreted formal first-order language. For example, it would maintain that ‘{ } ∈ {{ }}” is true in virtue of the set-theoretic fact that the empty set is a member of its singleton, and the semantic facts that ‘{ }’ designates the empty set,‘{{ }}’ designates its singleton, and ‘∈’ applies to an ordered pair of objects iff that pair's first component is a member of its second component.Now this theory may come so naturally as to seem trivial. My purpose here is to loosen its grip by “modelling” an alternative account of the alethic underpinnings of set-theoretic discourse. According to the Alternative theory, the point of having set-theoretic expressions (‘set’ and ‘∈’ will do) in a language is not to permit its speakers to talk about some special objects under a special relation; rather it is to clothe a higher-order language in lower-order garments.