简介：本章为玄宇宙计划原文的最后一章此序列文章内容将彻底完结（5/5）。
THE HYPERUNTVERSE PROGRAM 93
existential assertions concerning V. In fact,arguments aimed at showing
that the existence of large cardinals witnesses both the length of the ordinals and the thickness of power set, and are therefore faithful to the assumption that the universe is maximal, have been repeatedly given in the literature.¹⁶
Also forcing axioms have been advocated as “natural” axioms for set thcory,given their “maximizing” existential implications (see [2]).
In the Hyperuniverse Program.I is nowhereinvoked as an independently
existing well-determined reality. Nor is it called upon as a determined picture
of the universe forced upon us by our intuitions concerning sets. Instead,
V is intended as a meta-mathematical outcome. In saying this, we are not
thinking of the final result of a concluding process. What we have in mind is an ideal condition, which one can only better and better approximate. In the Hyperuniverse Program, V denotes the structure that satisfies whatever AB) set-thcorctic statements deserve to be regarded as true (either as a de facto or as a de jure set-theoretic truth). Le. the content of V, far from being understood in terms of a reality which is in se determined and which we should be faithful to when doing set theoryis meant to be a product of our own, progressively developing along with the advances of set theory, the development of the program with the resulting enrichment of the realm of
set-theoretic truth.
In particular, in the Hyperuniverse Program V plays the role of an out-
come that one can only approach by starting from the hyperuniverse as the
most suitable instantiation of the multiverse notion. The fact that within theHyperuniverse Program one endorses a multiverse perspective is explained as by Woodin, who says that “the refinements of Cohen's method of forcing in the decades since its initial discovery and the resulting plethora of prob-lems shown to be unsolvable, have in a practical sense almost compelled one to adopt” a multiverse position in contemporary set theory ([23). p.103).
Consider that, from a multiverse perspective,one works not with a unique
"system of all sets” but with many different ones, and deals with them as
meta-mathematical constructions, as models. From a multiverse perspective
one is thus naturally led to understand the expression “maximal properties
of the system of all sets” in terms of meta-mathematical features revealed by
comparing set-theoretic models. This is what is done in the Hypcruniversc
Program. Criteria like vertical and horizontal maximality are the rigorous
cxpression of what it means for an clement of the hyperuniverse, i.e., a count-able transitive model of ZFC, to display “maximum properties”, To put it in other terms, no need is seen in the Hyperuniverse Program to make existen-tial asscrtions concerning V in order to bc faithful to the idea that the system
of all sets must be maximal; in particular no need is seen to assume the ex-istence of large cardinals in the universe. Vice-versa, implications like those
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¹⁶See [16] for an extensive review of arguments given by set-theorists as to the faithfulness of large cardinals to maximality.
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See [16] for an extensive review of arguments given by set-theorists as to the faithfulness of large cardinals to maximality.
94 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN
of the Synthesis Conjecture concerning large cardinals are not seen as con- tradicting one's maximality expectations concerning the "system of all sets",
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.
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”.
But there are clear rebuttals to this argument. Consider, for instance, Lévy-
Shocnfield absoluteness,the absolutcness of Σ¹₂ statcments with respect to arbitrary outer models. This is provable in ZFC even if one allows arbitrary
real parameters. Extrapolation then naturally leads to Σ¹ʀ.absoluteness with arbitrary real parameters. But even Σ¹₃ absoluteness with arbitrary real
parameters is provably false. With arbitrary real parameters a consistent
principle can only be obtaincd by artificially taking "outer model” to mean
"set-generic outer model". As soon as one relaxes this to class-generic outer
models, the principle becomes inconsistent.
So, if one is so easily led to inconsistency when extrapolating from Σ¹₂ to Σ¹₃ absoluteness, how can one justify the extrapolation from Σ¹₁ measurability to projcctivemeasurability?
More reasonable would be the extrapolation with out parameters. Indeed,paramcter-free Σ¹₃ absoluteness, unlike the version
with arbitrary real parameters, is consistent with (and indeed follows from)the IMH.Thus a natural conclusion with regard to projective statements is the following: The principle of uniformity, which asserts that properties that hold for parameter-free projcctive sets also hold for arbitrary projective sets
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”.
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96 TATIANA ARRIGONT AND SY-DAVID FRIEDMAN
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