特殊篇章玄宇宙计划原文序列部分章 (10-5)

A possible objection to the above is that one uses large cardinals in V. rather than in inner models,to prove forms of the axiom of determinacy. such as PD, determinacy for all projective sets of reals. There are two common reasons given for asserting that PD is “true". One reason is based

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¹⁷Similar views as to the role of large cardinals in set theory are expressed by Shelah. See[19]. An opposing view, expressed in [23], is that the only basis for believing in the consistency of large cardinal axioms is believing in their truth in V. One can object, however, that Woodin's argument is based on a false analogy between large infinities and large finite sets It is true that the existence of large finite sets is implied by their consistency;this is simply because Vα has no proper inner modcls and therefore the existence of large finite sets is the same as their cxistence in inner models. This is obviously not the case with large infinities.

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THE HYPERUNIVERSE PROGRAM 95

on extrapolation: Since Borel and analytic sets are well-behaved (in the

sense that they are Lebesguc measurable and have the Baire and perfect set properties) and PD extends this to all projective sets, then PD must be “true”.

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