特殊篇章玄宇宙计划原文序列部分章 (13-5)

one recognizes ordinal-definable determinacy without real parameters, as oppposed to PD, as a de facto set-theoretic truth. This discussion of the ef-fects of the IMH on existing set-thcoretic developments also applies to other criteria for preferred universes that arise in the Hyperuniverse Program.

The plan of this paper is as follows.In Section 2 we describe the hyperuni-

verse and consider its relation to V.In Section 3 we introduce criteria for

preferred universes based upon principles of maximality and omniscience.

The current state of the HyperniverseProgram is summarized in Section 4.

while the final appendix is devoted to a broader discussionof maximal-

ity as well as to the roles of large cardinals and projective determinacy in

set-theoretic practice.

§2. The hyperuniverse. In contemporary set theory many methods are

available for creating new universes, i.e., models of ZFC,starting from

given ones: set-forcing.class-forcing.hyperclass-forcing (i.e., forcing whose

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　THE HYPERUNTVERSE FROGRAM 83

conditions are classes), ⁷ and model-thcoretic techniques.⁸ As a result. a

multitude of different universes are available to set-theorists. This abun-

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