1988/09/01 by Penelope Maddy · 3 citations
Mathematics · Economics, Econometrics and Finance · Computer Science · #Advanced Topology and Set Theory #Economic theories and models #Computability, Logic, AI Algorithms #Determinacy #Axiom of choice #Mathematics #Axiom #Property (philosophy) #Mathematical economics #Continuation #Natural number #Discrete mathematics #Set (abstract data type) #Set theory #Computer science #Epistemology #Philosophy
paper · doi:10.2307/2274569
openalex publication_date 1988/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
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 ω ω ). 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.