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Believing the axioms. II

Published online by Cambridge University Press: 12 March 2014

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

Extract

This is a continuation ofBelieving 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. IfA is such a set, we imagine an infinite gameG(A) between two players I and II. The players take turns choosing natural numbers. In the end, they have generated a real numberr (actually a member of the Baire spaceωω). Ifr is inA, I wins; otherwise, II wins. The setA 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.

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Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1988

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