A model of second-order arithmetic satisfying AC but not DC
Abstract
We show that there is a β-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a 12-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ZFC-. This work is a rediscovery by the first two authors of a result obtained by the third author.
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