An axiom schema of comprehension of zermelo–fraenkel–skolem set theory

History and Philosophy of Logic 11 (1):59-65 (1990)
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Unrestricted use of the axiom schema of comprehension, ?to every mathematically (or set-theoretically) describable property there corresponds the set of all mathematical (or set-theoretical) objects having that property?, leads to contradiction. In set theories of the Zermelo?Fraenkel?Skolem (ZFS) style suitable instances of the comprehension schema are chosen ad hoc as axioms, e.g.axioms which guarantee the existence of unions, intersections, pairs, subsets, empty set, power sets and replacement sets. It is demonstrated that a uniform syntactic description may be given of acceptable instances of the comprehension schema, which include all of the axioms mentioned, and which in their turn are theorems of the usual versions of ZFS set theory. Well then, shall we proceed as usual and begin by assuming the existence of a single essential nature or Form for every set of things which we call by the same name? Do you understand? (Plato, Republic X.596a6; cf. Cornford 1966, 317)



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Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1958 - Atlantic Highlands, NJ, USA: North-Holland.
From Mathematics to Philosophy.Hao Wang - 1974 - London and Boston: London.
Cantorian Set Theory and Limitation of Size.Joseph W. Dauben - 1988 - British Journal for the Philosophy of Science 39 (4):541-550.

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