David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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History and Philosophy of Logic 11 (1):59-65 (1990)
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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References found in this work BETA
Willard V. Quine (1963). Set Theory and its Logic. Harvard University Press.
John Mayberry (1986). Cantorian Set Theory and Limitation of Size. [REVIEW] Philosophical Quarterly 36 (144):429-434.
Francis Macdonald Cornford (1942). The Republic of Plato. Philosophical Review 51 (6):617-618.
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