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.

Restrictiveness Relative to Notions of Interpretation

“…The most prominent approach for this is to rule out theories because they restrict the development potential of the foundations of mathematics, i.e., violate the maxim MAXIMIZE Maddy (1988aMaddy ( ,1988bMaddy ( , 1998. Maddy (1998Maddy ( , 1997 gives a semi-formal account of restrictiveness by defining a corresponding formal notion based on a class of interpretations.…”

“…There is a tradition of naturalistic philosophy of set theory in which philosophers observe that some theories are proposed by set theorists as reasonable contenders for foundations of mathematics and others are not. The naturalistic philosopher aims to understand the reasons for these decisions of the mathematical experts, preferably on the basis of mathematical understanding of the theories involved.The most prominent approach for this is to rule out theories because they restrict the development potential of the foundations of mathematics, i.e., violate the maxim MAXIMIZE Maddy (1988aMaddy ( ,1988bMaddy ( , 1998. Maddy (1998Maddy ( , 1997 gives a semi-formal account of restrictiveness by defining a corresponding formal notion based on a class of interpretations.…”

The Argument of Mathematics

“…Penelope Maddy argues that the acceptability of axioms is itself a matter of reason though not of direct or deductive proof (Maddy, 1988). Hence, those practices may be judged atypical.…”

Towards a Theory of Mathematical Argument

“…Yet, turning to axiomatic systems -the very heart of what some may call the formal approach -there is, even there, room for informal logic. Penelope Maddy (1988) argues that the acceptability of axioms is itself a matter of reason though not of direct or deductive proof. She takes the development of set theory as a case study in this practice.…”