特殊篇章玄宇宙计划原文序列部分章 (12-10)

¹⁵Stronger forms of reflection lead to much larger cardinals. These are the principles in which the parameter is allowed to be a more complex object, such as a hyperclass(class of classes). hyperhyperclass(class of hyperclasses)....Carrying this out in the natural way leads quickly to inconsistency as Koellner has pointed out (see[15]), Carrying this out using the concept of embedding restores consistency and via work of Magidor (see[17] or [13].Theorem 23.6) leads to an cquivalence with the very large supercompact cardinals. However. it is not clear how to justify embedding reflection principles as unbiased or even natural principles of ordinal maximality.duc to the arbitrary nature of the embeddings involved (the relationship between A and its “reflected version" is given by an cmbedding with no uniqueness properties).

¹⁴ In particular the IMH is consistent with the regularity of all parameter-free definablc projective sets of reals. Allowing arbitrary real parameters makes a big diffcrence and con-

verts a principle compatible with the IMH to one which is not.

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All ne subjed to ISTU Ters ani C'ond furs

90 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

role of both large cardinal axioms and PD within set theory in the Appen-

dix.

What conclusion can be drawn as to justified criteria for preferred uni-

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