We present recent results on the model companions of set theory, placing them in the context of a 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 $H_{\kappa ^+}$ , as $\kappa $ 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). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). 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.

我们展示了关于集合论模型同伴的最新结果，将它们置于当前数学哲学辩论的背景下。我们首先描述模型陪伴概念对签名的依赖性，然后我们在集合论的特定情况下分析这种依赖性。我们认为，集合论最自然的模型伙伴描述（作为我们公理化集合论的签名变化） $H_{\kappa ^+}$ 的理论，因为 $\kappa$ 范围在无限基数之间。我们还挑出了 $2^{\aleph _0}=\aleph _2$ 作为连续统问题的唯一解，它可以（并且确实）属于集合论的某个模型伙伴（富含大基数公理）。在这样做的同时，我们揭示了由超紧性范围内的大基数公理丰富的集合论作为其模型伙伴（关于其在某些自然签名中的一阶公理化）$H_{\aleph_2}$ 的 理论 作为由 Woodin 公理 $(*)$ 的强形式给出（假定 $\mathsf {MM}^{++}$ 成立）。最后，根据受希尔伯特完备性公理启发的极大性形式，解释并证明了这种对集合论有效性的模型论方法。