特殊篇章玄宇宙计划原文序列部分章 (12-7)

tence of set models of ZFC does belong to the realm of de facto set-theoretic truth. The same applies to a weaker criterion inspired by a minimality prin-ciple, according to which one should prefer universes that satisfy the axiom of constructibility. V = L.Although the axiom of constructibility does allow for the existence of set models of ZFC (and more), it does not allow for the existence of inner models of ZFC with measurable cardinals. This

too stands in conflict with set-theoretic practice, i.e.,the existence of such

modcls belongs to the realm of de facto set-thcoretic truth (the point will be further discussed in the Appendix).

We turn now to the principle of maximality. A first point to make about

maximality is that one cannot have “structural maximality” within the hy-

peruniversc,in the sense that a preferred universe should contain all possible ordinals or real numbers. For there is no tallest countable transitive model of ZFC and over any such model new reals can be added to obtain another such model. What principle of maximaliry may be then imposed on elements of the hyperuniverse?

(Logical) Maximaliry:let be a variable that ranges over the elements of

the hyperuniverse. is (logically) maximalifall set-thcoretic statements with certain parameters which hold externally, i.e., in some universe containing ∪ as a "subuniverse",also hold internally, i.e..in some"subuniverse" of ∪.

Depending on what one takes as parameters and what onc takes for the

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