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

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 $Π21$-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)].

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Published: 21 September 2018