Wednesday, September 30, 2015

Infinite dependence regresses and set theory

Given the Axiom of Dependent Choice, the Axiom of Regularity in set theory is equivalent to the statement that there are no backwards infinite membership regresses, i.e., no cases where we have a backwards infinite sequence of sets ...,A−3,A−2,A−1,A-0, where each set is a member of the next. Why think this is true? Well, intuitively, a set depends on its members. That suggests that the reason to believe the Axiom of Regularity is that there cannot be an infinite dependency regress. And that in turn has all sorts of other consequences (including that there is a first cause).

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