特殊篇章玄宇宙计划原文序列部分章 (13-7)

work, where Woodin de facto regards the notions “generic-extension” and

"set-generic-extension” as synonymous, carlier work of the second author

of this paper led to the introduction of the class-generic multiverse around L

obtained by closing Lunder class-forcing and class-generic ground models.

as well as inner models of class-generic extensions that are not necessarily

themselves class-generic (see [5]).¹⁰ The set-generic multiverse and the class-generic multiverse are quite different:the former preserves large cardinals notions and does not lead beyond set forcing, whereas the latter can destroy large cardinals and leads to models that are not directly obtainable by class

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⁷See [6].

⁸Sce.eg.[14]. It is by manipulating universes of sets via these methods that mutually exclusive truth valucs can be assigned to set-theoretic sentences, thus proving them to be

independent from ZFC.

⁹The restriction to countable transitive models of ZFC is due to the fact that the existence of forcing extensions for such models can be proved in ZFC.

¹⁰Woodin explicitly rejects the possibility of considering a multiverse built on class forcing:

"there is no reasonable candidate for the definition of an expanded version of the set-generic

multiverse which allows class- forcing extensions and yet which preserves the existence of large

cardinals across the multiverse" ([23].p. 107). There are difficulties with Woodin's position.

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