The Ultrapower Axiom
The Ultrapower Axiom
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超能力公理
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
G. Goldberg
中科院分区：
文献类型：
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作者：
G. Goldberg
超紧基数的内模问题是现代集合论中的一个中心问题，它研究的是是否存在一个具有超紧基数的集合论典范模型。这个问题与更精确的强紧基数和超紧基数的等同性问题密切相关。本文通过引入一个在所有已知的集合论规范模型中都成立的超幂公理来抽象地探讨这两个问题。通过研究超幂公理在超紧基数假设下的结果，我们提供了内模问题可以解决的证据。在超幂公理下，强紧性与超紧性本质等价。
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that there is a supercompact cardinal, we provide evidence that the inner model problem can be solved. Moreover, we establish that under the Ultrapower Axiom, strong compactness and supercompactness are essentially equivalent.