特殊篇章玄宇宙计划原文序列部分章 (12-11)

verses? We have so far formulated two candidate criteria: ordinal maxi-mality and power set maximality. The ideal situation would be to combine

them into a singlc consistent critcrion. i.e., a criterion that is satisficd by at

least one element of the hyperuniverse. This is not trivial, since power set

maximality and ordinal maximality contradict cach other. One is thus led

to the following conjecture:

SYNTHESIS CONJECTURE. Let power set maximality* (IMH*) be power set AB) maximality(IMH) restricted to ordinal maximal universes (ie.,the state-ment that if a sentence holds in an ordinal maximal outer model of ∪ then it holds in an inner model of ∪). Then the conjunction of power set maximal-ity* (IMH*) and ordinal maximality is consistent. l.e., there are universes which simultancously satisfy both criteria.

A proof of the Synthesis conjecture is within reach, as it only dcmands the

existing method for proving the consistency of the IMH (see [8])together

with a careful understanding of how Jensen coding can be done in the

presence of small large cardinal properties. Via the Hyperuniverse Program

the Synthesis Conjecture is effective in yielding new (first-order) set-theoretic

axioms, including solutions to independent questions. As universes which

witness the Synthesis Conjecture (i.e., which arc ordinal maximal and satisfy

IMH*) are preferred universes, first-order properties shared by all such

universes are true in I and may be adopted as new axioms. Examples of

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