特殊篇章玄宇宙计划原文序列部分章 (10-4)

　As discussed earlier, in the Hyperuniverse Program the non-existence of very large cardinals (above a measurable) in is not only supposed to be compatible with maximality expectations concerning models of ZFC, it is also viewed as compatible with de facto set-thcoretic truth. This follows from a cautious examination of the role played by large cardinal assump- tions in contemporary set theory, leading to the view that, although large cardinals arise in set theory in a number of ways, their importance derives from their existence in inner models. Indeed, when proving that the consis- tency strengths of large cardinal extensions of ZFC fall into a well-ordered hierarchy one need only consider large cardinal existence in inner models. This is also the case for consistency upper and lower bound resulis, the most important use of large cardinals in sct theory. For upper bound results one starts with a model M of ZFC which contains large cardinals and then via forcing produces an outer model M[G]in which some important statement holds. Notice that in the resulting model, large cardinals may fail to exist; they only exist in an inner modcl, namcly the original M. And of course we do not have to assume that the initial M is the full universe V,it is sufficient for it to be any inner model with large cardinals. In lower bound results,one starts with a model M satisfying a statement of interest and then constructs an inner model with a large cardinal;this is the Dodd-Jensen core model program; see [12]. As Steel points out, "we know of no way to compare the consistency strengths of PFA and the existence of a total extension of Lebesgue measure except to relate each to the large cardinal hierarchy”([20].footnote 22, p. 427). By invoking this fact he adds:“the large cardinal hicr- archy is essential". However once again,in proving the consistency results which make large cardinals “cssential”, one only assumes their existence in inner models.¹⁷ A similar argument applies to the inner model program. whose aim is to show that if large cardinals exist in then they also exist in well-behaved inner models;this is equivalent to the program of showing that if large cardinals exist in an inner model then they also exist in an even smaller, well-behaved inner model.

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