特殊篇章玄宇宙计划原文序列部分章 (5-2)

§4. Conclusions. The Hyperuniverse Program presented in this paper is a

new approach to set-thcoretic truth, aimed at enlarging the realm oftrue-in- I

statements beyond ZFC. To this purpose the program develops a justifiable strategy, and regards the intrinsic reasonableness of this strategy as a guar-antee for the truth of the results obtained. More precisely, one introduces the hyperuniverse as the most suitable realization of the multiverseconcept and puts it to work for the purpose of comparing different pictures of the set-theoretic universe (countable transitive models of ZFC)in light of crite-ria for prefcrring some universes over others. First-order properties shared by all preferred universes are taken to be true in V.By invoking the criteria of ordinal (vertical) maximality and power sct (horizontal) maximality, a suitable realization of the program is obtained. By postulating the exis-tence of an element of the hyperuniversc that satisfies the natural synthesis of these criteria (i.e., the Synthesis Conjecture), one arrives at statements which are true in V yet independent from ZFC. These statements contradict the existence of very large cardinals but are consistent with their existence in inner models, and they contradict projective determinacy but are consistent

with determinacy for sets of reals which are ordinal-definable without real

parameters. This leads to a reassessment of the roles of large cardinals and determinacy in sct theory.

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