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Believing the axioms. II

  • Penelope Maddy (a1)

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.

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J. W. Addison and Y. N. Moschovakis [1968] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 708712.

J. Barwise , editor [1977] Handbook of mathematical logic, North-Holland, Amsterdam, 1977.

D. Blackwell [1967] Infinite games and analytic sets, Proceedings of the National Academy of Sciences of the United States of America, vol. 58 (1967), pp. 18361837.

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R. M. Solovay [1969] The cardinality of Σ21 sets, Foundations of Mathematics ( J. J. Bulloff et al., editors), Symposium papers commemorating the sixtieth birthday of Kurt Gödel, Springer-Verlag, Berlin, 1969, pp. 5873.

R. M. Solovay , W. N. Reinhardt , and A. Kanamori [1978] Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.

M. Wilson [1979] Maxwell's condition—Goodman's problem, British Journal for the Philosophy of Science, vol. 30 (1979), pp. 107123.

P. Wolfe [1955] On the strict determinacy of certain infinite games, Pacific Journal of Mathematics, vol. 5 (1955), pp. 841847.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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