特殊篇章玄宇宙计划原文序列部分章 (12-8)

concept of “subuniverse”, different criteria for maximal universes arise from(and are justified in light of) this principle.Here are two examples.

• Criterion of ordinal (or vertical) maximaliry:this criterion appeals to

maximality with respect to the ordinals, where models have fixed the

power-set operation. Let us define a universe w to be a lengthening of

v ifv is a (proper) rank initial segment ofw. is ordinal maximal iff it

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THE HYPERUNIVERSE PROGRAM 89

has a lengthening w such that for all first-order formulas φ and subsets A of ∪ belonging to ω, if φ(A ∩ ∪α ) holds in ∪ᵦ then e(A n va) holds in ua for some pair of ordinals α ＜ β in ∪α (where ta denotes the collection of sets in e of rank less than α).

• Criterion of power set (or horizonial) maximaliry: this criterion appeals to maximality with respect to power set.where models have fixed ordi- nals. If a parameter-free sentence holds in some outer model of ∪ (i.e., in some universe w containing ∪ with the same ordinals as ∪), then it holds in some inner model of ∪ (i.e., in some universe to contained in ∪ with the same ordinals as ∪).

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