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A model of second-order arithmetic satisfying AC but not DC
Journal of Mathematical Logic ( IF 0.9 ) Pub Date : 2018-08-28 , DOI: 10.1142/s0219061318500137
Sy-David Friedman 1 , Victoria Gitman 2 , Vladimir Kanovei 3
Affiliation  

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-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 [Formula: see text]. 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)].

中文翻译:

满足交流但不满足直流的二阶算术模型

我们证明了存在选择方案成立的二阶算术[公式:参见文本]-模型,但是对于[公式:参见文本]-断言,从属选择方案失败,证实了斯蒂芬辛普森的猜想。作为一个推论,我们得到反射原理,说明每个公式都反映到一个传递集,在[公式:见文本]的模型中可能会失败。这项工作是前两位作者对第三位作者在 [VG Kanovei,关于选择的可数公理的描述形式,非经典逻辑和集合论的调查,Work Collect.,莫斯科,3-136 (1979)]。
更新日期:2018-08-28
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