特殊篇章玄宇宙计划原文序列部分章 (12-5)

Criteria of this kind reflect the interests of specific groups of set-thcorists or mathematicians, As a result, there may be as many different such criteria as there are areas of set theory or mathematics. Moreover, as interests in set thcory change, so may these criteria. Thus at the outsct, no sclection of universes can be made according to them that can presume to be universally recognized as lcgitimate within the set-thcoretic community as a whole. Is there a better way to select preferred universes?

The positive answer to this question given by the Hyperuniverse Program is that by focusing instead only on the most general features of the hype- runiverse and formulating principles based upon them,one is capable of suggesting (and justifying) criteria for preferred universes. This is based on the obvious fact that the hyperuniverse consists of ZFC models that may be

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¹¹See.e.g.[19) on the advantages of assuming GCH as an axiom.

¹²One may add to the list Woodin's axiomatic proposals and conjectures, introduced in

[22], based on Ω-logic. The latter is a logic which can be proved (under the assumption of the existence of a proper class of Woodin cardinals) to be unaffocted by sct-forcing. But as discussed earlier, one cannot justify an exclusive focus on set-forcing in suggesting new axioms and conjectures.

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88 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

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