整体性公理与V= HOD
The Wholeness Axioms and V=HOD
摘要：
Paul Corazza提出的整体性公理化了初等嵌入j的存在性：V--> V。通过用一元函数符号j来表示嵌入来扩充集合论的常用语言，它们通过将基础理论ZFC限制为集合论的常用语言来避免Kunen不一致。因此，在整体性公理下，人们不能像库南在他著名的不一致性证明中那样，在有j的语言中诉诸替换公理。事实上，很容易看出，整体性公理具有严格低于I_3基数存在的一致性强度。本文证明了如果整体性公理WA_0本身是相容的，那么它与V=HOD是相容的。证明的一个结果是，各种整体性公理WA_n并不都是等价的。此外，理论ZFC+WA_0是可公理化的。
英文摘要：
The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding j:V-->V. Formalized by augmenting the usual language of set theory with an additional unary function symbol j to represent the embedding, they avoid the Kunen inconsistency by restricting the base theory ZFC to the usual language of set theory. Thus, under the Wholeness Axioms one cannot appeal to the Replacement Axiom in the language with j as Kunen does in his famous inconsistency proof. Indeed, it is easy to see that the Wholeness Axioms have a consistency strength strictly below the existence of an I_3 cardinal. In this paper, I prove that if the Wholeness Axiom WA_0 is itself consistent, then it is consistent with V=HOD. A consequence of the proof is that the various Wholeness Axioms WA_n are not all equivalent. Furthermore, the theory ZFC+WA_0 is finitely axiomatizable.
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