偶数和库南矛盾
Even ordinals and the Kunen inconsistency
摘要：
本文在不假定选择公理的情况下，对超越Kunen不一致性的大基数理论或无选择大基数公理作出了贡献。本文的第一部分研究了一个周期性现象：假设无选择的大基数公理，累积层次的性质变成交替的偶数和奇数的行列。本文的第二部分探讨了超滤的结构下选择的大基数公理，利用这些公理意味着作者的超幂公理的弱形式。本文的第三部分也是最后一部分研究了无选择大基数的一致性强度，包括证明了假设DC，从$V_{λ+3}$到$V_{λ+3}$的初等嵌入的存在暗示了ZFC + $I_0$的一致性。根据Schlutzenberg最近的一个结果，从$V_{λ+2}$到$V_{λ+2}$的初等嵌入是不够的。
英文摘要：
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding from $V_{λ+3}$ to $V_{λ+3}$ implies the consistency of ZFC + $I_0$. By a recent result of Schlutzenberg, an elementary embedding from $V_{λ+2}$ to $V_{λ+2}$ does not suffice.
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