特殊篇章玄宇宙计划原文序列部分章 (10-1)

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THE HYPERUNTVERSE PROGRAM 93

existential assertions concerning V. In fact,arguments aimed at showing

that the existence of large cardinals witnesses both the length of the ordinals and the thickness of power set, and are therefore faithful to the assumption that the universe is maximal, have been repeatedly given in the literature.¹⁶

Also forcing axioms have been advocated as “natural” axioms for set thcory,given their “maximizing” existential implications (see [2]).

In the Hyperuniverse Program.I is nowhereinvoked as an independently

existing well-determined reality. Nor is it called upon as a determined picture

of the universe forced upon us by our intuitions concerning sets. Instead,

V is intended as a meta-mathematical outcome. In saying this, we are not

thinking of the final result of a concluding process. What we have in mind is an ideal condition, which one can only better and better approximate. In the Hyperuniverse Program, V denotes the structure that satisfies whatever AB) set-thcorctic statements deserve to be regarded as true (either as a de facto or as a de jure set-theoretic truth). Le. the content of V, far from being understood in terms of a reality which is in se determined and which we should be faithful to when doing set theoryis meant to be a product of our own, progressively developing along with the advances of set theory, the development of the program with the resulting enrichment of the realm of

set-theoretic truth.

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