Home
• Aa
• Aa

# Believing the axioms. II

Abstract

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.

Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

J. E. Fenstad [1971] The axiom of determinateness, Proceedings of the second Scandinavian logic symposium (J. F. Fenstad , editor), North-Holland, Amsterdam, 1971, pp. 4161.

H. Field [1985] On conservativeness and incompleteness, Journal of Philosophy, vol. 82 (1985), pp. 239260.

A. Levy and R. M. Solovay [1967] Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.

P. Maddy [1980] Perception and mathematical intuition, Philosophical Review vol. 89 (1980), pp. 163196.

D. A. Martin [1968] The axiom of determinateness and reduction principles in the analytic hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.

D. A. Martin [1975] Borel determinacy, Annals of Mathematics, ser. 2, vol. 102 (1975), pp. 363371.

M. Resnik [1981] Mathematics and a science of patterns.: ontology and reference, Noûs, vol. 15 (1981), pp. 529550.

M. Resnik [1982] Mathematics as a science of patterns: epistemology, Noûs, vol. 16 (1982), pp. 95105.

S. Shapiro [1983] Mathematics and reality, Philosophy of Science, vol. 50 (1983), pp. 523548.

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.

Recommend this journal

The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic