哥白尼的集合多重宇宙
The Copernican Multiverse of Sets
摘要：
我们开发了一个非类型化的框架的多重宇宙的集合论。利用新的符号$\mathsf{Uni}(\mathcal{U})$和$\mathsf{Mod}(\mathcal{U, σ})$，用语义动机公理扩展$\mathsf{ZF}$，分别表示$\mathcal{U}$是论域和$σ$在论域$\mathcal{U}$中为真。在这里，$σ$的范围超过了增强语言，导致说谎风格的现象进行分析。该框架既与广泛的多元宇宙概念兼容，又提出了自己的语义和语义动机的多元宇宙原则。特别是，该框架与必然性的演绎规则紧密相连，该规则表示多元宇宙理论只能证明它也证明在所有宇宙中成立的陈述。我们认为，这可能被认为是一个哥白尼的原则，背景理论并不持有优先地位的理论，其内部的宇宙。我们的主要数学结果是一个引理封装技术本地解释各种各样的扩展我们的基本框架，在更熟悉的理论。我们应用这一点来表明，对于一系列这样的语义动机的扩展，它们的一致性强度最多略高于基本理论$\mathsf{ZF}$，因此不会严重限制集合论多元宇宙的多样性。最后，我们的案例研究应用框架的两个多元宇宙概念的集合论：算术绝对性和乔尔D。哈金斯的多重宇宙理论
英文摘要：
We develop an untyped framework for the multiverse of set theory. $\mathsf{ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf{Uni}(\mathcal{U})$ and $\mathsf{Mod}(\mathcal{U, σ})$, expressing that $\mathcal{U}$ is a universe and that $σ$ is true in the universe $\mathcal{U}$, respectively. Here $σ$ ranges over the augmented language, leading to liar-style phenomena that are analysed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf{ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins' multiverse theory.
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