特殊篇章玄宇宙计划原文序列部分章 (10-7)

is false. Thus the regularity of projective sets is a reasonable extrapolation from the regularity of Borel and analytic sets provided one does not allow parameters. Indeed,parameter-free PD(or even ordinal-definable determi-nacy without real parameters) and the existence of inner models with very large cardinals are consistent with the IMH (and very likely with a witness to the Synthesis Conjecture), but PD with paramcters and the cxistence of inner

models with very large cardinals containing an arbitrarily given real are not.

A sccond rcason for asscrting the “truth” of PD is that it “settles all natural

questions about HC (the set of hereditarily countable sets)". This assertion

is based on the fact that assuming large cardinals, you cannot changc thc

first-order theory of HC by set-forcing and this theory is in some sense

described by PD. But this ignores the fact the theory of HC can changc,

even at the least possible level (Σ¹₃) if one allows other ways of enlarging

the universe, even ways which preserve the existence of very large cardinals.

And there are simple examples of such statements (such as the existencc

of modcls of very large cardinals with a small amount of “iterability”)

The conclusions reached by the Hyperuniverse Program through the use of maximality principles also yicld strong conclusions about the theory of HC(conflicting with PD), but without any need to refer to “set-forcing”.

REFERENCES

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