集合论的模型伴随谱、类属绝对性和连续统问题
The model-companionship spectrum of set theory, generic absoluteness, and the Continuum problem
摘要：
我们发现，$Π_2$ -属性的二阶或三阶算术正式在适当的自然签名的强制性明显较弱的概念重叠的标准概念的一致性（假设大基数公理）。在这样的$Π_2$ -属性，我们提到：否定的连续统假设，苏斯林假设，否定怀特海猜想的自由群，不存在的外部自同构的卡尔金代数等.特别是，这给出了一个后验的解释，成功迫使（和迫使公理）在生产模型，这些属性。我们的主要结果涉及通用的绝对性定理二阶算术，Woodin的公理$(*)$和迫使公理罗宾逊的概念模型伴侣（适用于集合论）。我们还简要概述了这些结果以何种方式提供了一个论点，以反驳连续统假设。
英文摘要：
We show that for $Π_2$-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal axioms). Among such $Π_2$-properties we mention: the negation of the Continuum hypothesis, Souslin Hypothesis, the negation of Whitehead's conjecture on free groups, the non-existence of outer automorphisms for the Calkin algebra, etc... In particular this gives an a posteriori explanation of the success forcing (and forcing axioms) met in producing models of such properties. Our main results relate generic absoluteness theorems for second order arithmetic, Woodin's axiom $(*)$ and forcing axioms to Robinson's notion of model companionship (as applied to set theory). We also briefly outline in which ways these results provide an argument to refute the Continuum hypothesis.
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