Where do sets come from?

Publisher: Cambridge University Press

E-ISSN: 1943-5886|56|1|150-175

ISSN: 0022-4812

Source: The Journal of Symbolic Logic, Vol.56, Iss.1, 1991-03, pp. : 150-175

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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.