特殊篇章玄宇宙计划原文序列部分章 (12-4)

In Section I we made the point that by subscribing to the Hyperuniverse Program one is expected to conform to principles and criteria for preferred universes that arise from an unbiased look at the hyperuniverse, so as to obtain a selection of universes that is justifiable. The program thus excludes the possibility that needs arising from specific areas of set-theoretic or math- cmatical practice play a role in formulating criteria for preferred universes. Therefore statements to the effect that one should prefer universes in which principles hold that resolve the difficulties arising in a specific area of set theory or mathematics are not candidates for such criteria. Let us give some examples of such non-criteria.

a. The generalised continuum hypothesis (GCH), which is very effective in resolving a wide range of questions in set theory; ¹¹

b.V=L.a theory which yields a powerful infinitary combinatorics which can be used to resolve even more problems in set theory than GCH;

jcctive Determinacy(PD),which yiclds an attractive theory of pro- jective sets of reals;

d. Forcing axioms (such as MA, BPFA, BMM), which, like V = L.have great combinatorial strength. ¹²

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