论文标题
在弱设定理论中的双重解释
Bi-interpretation in weak set theories
论文作者
论文摘要
与集合理论中强大的相互可解释性现象相反，阿里·埃纳亚特（Ali Enayat）证明了双重释放是不存在的：扩展ZF的不同理论永远是双重释放的，并且ZF模型仅在同构时才可互隔离。然而，对于自然弱设定的理论，我们证明了包括zermelo-fraenkel set理论$ \ text {zfc}^ - 无功率集和zermelo set理论z的$，存在非琐事的双性化实例。具体而言，有$ \ text {zfc}^ - $具有良好的型号，但不是同构的，但不是同构的---甚至$ \ langle h_ {ω_1}，\ in \ in \ rangle $ and $ \ langle h_____________________________________________________________________________-理论扩展了$ \ text {zfc}^ - $。同样，使用Mathias的构造，我们证明了ZF的每个模型都可以通过Zermelo Set理论模型进行双重解剖，其中替换公理会失败。
In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory $\text{ZFC}^-$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\text{ZFC}^-$ that are bi-interpretable, but not isomorphic---even $\langle H_{ω_1},\in\rangle$ and $\langle H_{ω_2},\in\rangle$ can be bi-interpretable---and there are distinct bi-interpretable theories extending $\text{ZFC}^-$. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.
