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Believing the axioms. II

Published online by Cambridge University Press:  12 March 2014

Penelope Maddy
Penelope Maddy*
Affiliation:
Department of Philosophy, University of California at Irvine, Irvine, California 92717
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Extract

This is a continuation of Believing the axioms. I, in which nondemonstrative arguments for and against the axioms of ZFC, the continuum hypothesis, small large cardinals and measurable cardinals were discussed. I turn now to determinacy hypotheses and large large cardinals, and conclude with some philosophical remarks.

Determinacy is a property of sets of reals. If A is such a set, we imagine an infinite game G(A) between two players I and II. The players take turns choosing natural numbers. In the end, they have generated a real number r (actually a member of the Baire space ωω). If r is in A, I wins; otherwise, II wins. The set A is said to be determined if one player or the other has a winning strategy (that is, a function from finite sequences of natural numbers to natural numbers that guarantees the player a win if he uses it to decide his moves).

Determinacy is a “regularity” property (see Martin [1977, p. 807]), a property of well-behaved sets, that implies the more familiar regularity properties like Lebesgue measurability, the Baire property (see Mycielski [1964] and [1966], and Mycielski and Swierczkowski [1964]), and the perfect subset property (Davis [1964]). Infinitary games were first considered by the Polish descriptive set theorists Mazur and Banach in the mid-30s; Gale and Stewart [1953] introduced them into the literature, proving that open sets are determined and that the axiom of choice can be used to construct an undetermined set.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 736 - 764
Copyright
Copyright © Association for Symbolic Logic 1988

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