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arXiv · 1808.04732

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

Sy-David Friedman·Victoria Gitman·Vladimir Kanovei
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Abstract

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 ${\rm ZFC}^-$. This work is a rediscovery by the first two authors of a result obtained by the third author.

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Sy-David Friedman, Victoria Gitman, Vladimir Kanovei. 2018-08-14. A model of second-order arithmetic satisfying AC but not DC. https://arxiv.org/abs/1808.04732

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