论文标题
关于连续性问题的模型陪伴可以说什么
What model companionship can say about the Continuum problem
论文作者
论文摘要
我们介绍了有关集合理论的模型伴侣的最新结果，将它们置于数学哲学中当前辩论的背景下。我们首先描述模型伴侣身份对签名的概念的依赖性，然后在集合理论的特定情况下分析了这种依赖性。我们认为，集合理论的最自然模型伴侣将$ h_ {κ^+} $的理论描述为$κ$范围的$ h_ {κ^+} $都会变化）。我们还将$ 2^{\ Aleph_0} = \ Aleph_2 $作为连续问题的唯一解决方案，可以（并且确实）属于SET理论的某些模型伴侣（富含大型的基本公理）。最后，通过受希尔伯特完整性的公理启发的最大形式来解释和合理地解释和合理的这种模型理论理论方法。
We present recent results on the model companions of set theory, placing them in the context of the current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{κ^+}$, as $κ$ ranges among the infinite cardinals. We also single out $2^{\aleph_0}=\aleph_2$ as the unique solution of the Continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert's axiom of completeness.
