A model of second-order arithmetic satisfying AC but not DC

摘要

We show that there is a beta-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a Pi(1)(2)-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 in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 (1979)].