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THE HYPERUNIVERSE PROGRAM 81

modify the procedures adopted in order to integrate them with other, equally

reasonable procedures.⁵

In short, formulating de jure set-theoretic truth, which lies at thecore of theHyperuniverse Program, may be understood as the active response of a non-Platonistically minded mathematician, who believes that it makes sense to search for new truths in V beyond de facto sct-thcoretic truths. This contrasts

with any form of skepticism concerning such a search, be it motivated by

the assumption that such a scarch is hopcless or by the confidence, possibly

grounded on Platonism,that whatever the well-determined features of V

are, they will somchow manifest themsclves without any cffort of our own.

Equivalently, one may characterize the Hyperuniverse Program as a dynamic

approach to set-theoretic truth,freefrom external constraints(although

internally regulated), in contrast to any static Platonistic view that truth

conccrning scts is restricted to a fixed state of affairs to which onc must be

“faithful”.

The stance of the advocate of the Hypcruniverse Program towards exist-

ing set-theoretic developments is both complex and surprising. Of course

thc latter cxplicitly cnter into the program insofar as onc aims at obtain- AB ing preferred universes that,beyond conforming to certain criteria and not

contradicting existing de facto set-thcoretic truth, are successful in deciding

independent questions. Moreover the techniques needed to establish the

cxistencc of preferred universes are provided by existing developments in

set theory or by new developments inspired by the program which extend

existing developments. There is another reason, however, for the Hyperuni-verse Program to explicitly call upon set-theoretic developments, albeit in ancgative way. When declaring the intention of extending ZFC so as to settlc independent questions, one also requires that one be as nbiased as possiblc

as to the way such questions should be settled and as to which principles

and criteria for preferred universes onc should formulate. In particular, the

latter must not be chosen at the outset so as to be apt for settling questions

independent of ZFC, or for mecting the necds of some particular arca of

existing set-theoretic practice.Nor should specific mathematical hypotheses

be invoked in formulating such principles and criteria (c.g., large cardinal

or forcing axioms). The rationale bchind unbiasedness is twofold. On the

one hand, one wants to be as cautious as possible as to what sct-thcoretic

developments beyond ZFC belong to the realm of de facto set-theoretic

truth.Endorsing this attitude mcans doing justice to the fact that disparate

views have been advanced within the set-theoretic community on this mat-

ter. Secondly, one aims at formulating principles and criteria starting from

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　 ⁵An example of this will be given in Section 3, where the criterion of power set mapcimaliry

is modificd so as to be compatible with the criterion of ordinal maximality.

⁶See,e.g..[22] and [19] for different positions on the status of large cardinal axioms or thc

axiom of detcrminacy AD˪[ʀ] in contemporary set theory.

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82 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

an analysis of the hyperuniverse which focus exclusively on its most general

features. As a result, the principles chosen and the criteria derived from

them are expected to yield a justified selection of preferred universes on the

solc basis of onc's acquaintancc with the most basic aspects of set thcory.

A surprise is that,unbiasedness notwithstanding, the Hyperuniverse Pro-gram leads to results that strongly affect our understanding of the corpus

of already existing set-theoretic developments. This is the case, eg., if one

adopts the Inner Model Hypothesis (IMH), as formulated in [7]1, as a criterionfor preferred universes, providing a suitable description of what it means for a countable transitive modcl of ZFC to be maximal (fixing the ordinals).

This hypothesis settles many questions independent of ZFC, but also has

implications of a revisionary character with respect to what is sometimes

assumed without question by the set theory community: although the IMH

is compatible with the internal consistency of very large cardinals (ie., theirexistence in inner models), it contradicts their existence in the universe V as a AB whole. This may be regarded as disruptive, providing evidence contra rather

than pro the hypothesis. By taking it seriously, however, one may nonethe-

less come to the unexpected conclusion that the IMH does not contradict the practice of set theory after all, as it is the existence of large cardinals

in inner models, and not in V.that has gained the status of an ultimate,

unrevisable assumption in set theory, one which we are constrained not to

contradict in proposing new axioms. In other words, one recognizes the

internal consistency of large cardinals, as opposed to their actual existence

in the universe, as a de facto set-theoretic truth. An analagous phenomenon

regards projective determinacy (PD): the IMH contradicts PD but is consis-

tent with the determinacy of sets of reals which are ordinal-definable without real parameters. Thus the IMH violates the principle of uniformity,which

asserts that natural projective statements relativise to real parameters, and

one recognizes ordinal-definable determinacy without real parameters, as oppposed to PD, as a de facto set-theoretic truth. This discussion of the ef-fects of the IMH on existing set-thcoretic developments also applies to other criteria for preferred universes that arise in the Hyperuniverse Program.

The plan of this paper is as follows.In Section 2 we describe the hyperuni-

verse and consider its relation to V.In Section 3 we introduce criteria for

preferred universes based upon principles of maximality and omniscience.

The current state of the HyperniverseProgram is summarized in Section 4.

while the final appendix is devoted to a broader discussionof maximal-

ity as well as to the roles of large cardinals and projective determinacy in

set-theoretic practice.

§2. The hyperuniverse. In contemporary set theory many methods are

available for creating new universes, i.e., models of ZFC,starting from

given ones: set-forcing.class-forcing.hyperclass-forcing (i.e., forcing whose

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　THE HYPERUNTVERSE FROGRAM 83

conditions are classes), ⁷ and model-thcoretic techniques.⁸ As a result. a

multitude of different universes are available to set-theorists. This abun-

dance of ZFC models has recently led to the introduction of the multiverse

as a new set-theoretic notion, and to related discussions about whether the

multiverse may represent the proper starting point foraddressing questions

concerning truth in set theory. Depending on one's view as to which ZFC models should enter into it, quite different pictures of the multiverse have been suggested in the literature. Diverging views have been expressed as well as to how the multiverse may work as a proper framework for pronouncing

on matters of set-theoretic truth. In this section we will review existing al-

ternative proposals concerning the multiverse and present the hyperuniverse

as an optimal realization of the multiverse concept.

Both Woodin and the second author have used the term multiverse for

collections of universes obtained from one or more initial models of ZFC ) via some method for manipulating them. In particular, in [23] Woodin starts

from countable transitive models M of ZFC, and takes the multiverse around

M to be the collcction gencrated by closing under set-generic extensions

and set-generic ground models (this is what Woodin calls the (ser-)generic

multiverse gencrated from M).⁹ Also V is regarded by Woodin so that a

(set-)generic multiverse may be generated from it.To this purpose one con-siders (set-)gencric extensions as Boolean valued models, i.e., models having

the form V ᴮ,where B is a complete Boolean algebra. In contrast to this

work, where Woodin de facto regards the notions “generic-extension” and

"set-generic-extension” as synonymous, carlier work of the second author

of this paper led to the introduction of the class-generic multiverse around L

obtained by closing Lunder class-forcing and class-generic ground models.

as well as inner models of class-generic extensions that are not necessarily

themselves class-generic (see [5]).¹⁰ The set-generic multiverse and the class-generic multiverse are quite different:the former preserves large cardinals notions and does not lead beyond set forcing, whereas the latter can destroy large cardinals and leads to models that are not directly obtainable by class

＿＿＿＿＿＿　　

⁷See [6].

⁸Sce.eg.[14]. It is by manipulating universes of sets via these methods that mutually exclusive truth valucs can be assigned to set-theoretic sentences, thus proving them to be

independent from ZFC.

⁹The restriction to countable transitive models of ZFC is due to the fact that the existence of forcing extensions for such models can be proved in ZFC.

¹⁰Woodin explicitly rejects the possibility of considering a multiverse built on class forcing:

"there is no reasonable candidate for the definition of an expanded version of the set-generic

multiverse which allows class- forcing extensions and yet which preserves the existence of large

cardinals across the multiverse" ([23].p. 107). There are difficulties with Woodin's position.

Class-forcing and hyperclass-forcing have the same status as sel-forcing within set theory.

Moreover, by restricting to set-forcing and assuming the existence of large cardinals, one

artificially avoids a real difficulty of any unbiascd scarch for new set-thcoretic axioms, that of dealing with justifiable principles that may destroy large cardinal axioms.

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84 TATIANA ARRIGONIAND SY-DAVID FRIEDMAN

forcing. Also Hamkins has recently formulated a view of the multiverse,

apparently dissociating from both Woodin and the second author. What in

[11] is referred to as the multiverse is, in fact, not a collection of ZFC mod-

cls that can be gencrated from initial universes by closing under specified

procedures. Rather,the multiverse is described as the multitudeconsist-

ing of all set-thcorctic universes that have been constructed so far and may

be produced in the future, possibly including non well-founded models and

models of systems other than ZFC. The result is a heterogencous open-ended

plurality, of which no overall unified description can be given.

Nor is there consensus among set-theorists as to whether and how the

multiverse should be regarded as a context for determining questions of

truth. Hamkins' proposal of a heterogencous open-ended multiversc, for

instance, is accompanied by the twofold invitation to abandon the “dream

solution template for the CH”, according to which the truth value of CH

must be decided by some new axiom for set theory, and to consider the

question whether CH holds or not as already definitively settled, as the result

of our knowledge of the different truth values it may assume in the different

universes of the multiverse. In (23), instead, the set-generic multiverse is

introduced in order to scrutinize the set-generic multiverse conception of

truth. According to the latter, a sentence formulated in the language of set

theory is true if it holds absolutely in the multiverse generated by V.ie., ifit

holds in cach universe belonging to that multiverse. Were one to adopt the AB) set-generic multiverse conception of truth, one should declare that a sentence

like CH lacks truth value. This is not Woodin's conclusion, however. In fact

he argues that the set-generic multiverse conception of truth is untenable

because it violates principles that he regards as essential for any notion of

truth for the set-theoretic universe (sce [23]).

However note that, despite Woodin's and Hamkins' different mathemati-

cal understanding of the multiverse, and their diverging positions as to the

status of sentences independent of ZFC, there is a point at which their vicws of the multiverse are more similar than may appear at the outset.In consid-cting whether byinvokingthemultiverseeone may introduce a suitable notion of truth for sct-theorctic sentences, both Woodin and Hamkins tacitly start from the assumptionthatone shouldregard the multiverseas anultimate pluralityofZFCmodeisthatcannot .reduced to some-thingthat is more fund: Asaresult.theyarebothledtocandidates

for a notion of set-thcoretic truth that are highly incomplete,allowingscn-tences of set theory which are ncither true nor falsc. This assumption, shared by Woodin and Hamkins, is worth emphasizing as it is clearly reiccted by the HyperuniverscProgram(seeDesideratum 2 below),which we now present as a distinctive approach to set-theoretic truth making use of the multiverse concept.

The Hyperuniverse Program may be understood as an attempt to arrive at new de jure sct-theoretic truths by starting from a picture of the multiverse.

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THE HYPERUNIVERSE PROGRAM 85

　that faithfully summarizes the full plethora of results obtainable in contem-porary sct theory. That one focuses on well-founded models of ZFC when using this approach just amounts to expressing the twofold conviction that the axioms of ZFC are de facto set-thcoretic truths and that it is only the well-founded models of this theory that provide plausible pictures of the set-thcoretic universe. The Hyperuniverse Program thus begins by asserting thatthe multiverse should satisfy a maximality and a well-definedness criterionthat only the collcction of all countable transitive models of ZFC can mect.

More precisely:

DESIDERATUM 1.The multiverse should be as rich as possible but it should not be an ill-defined or open-ended multiplicity.

In stating this, onc has two aims. First, onc is motivated by the fact that

the methods for creating well-founded universes existing in contemporary

set thcory go well beyond set-forcing or class-forcing (hence the multivcrsc

should include more than set- or class-generic extensions and ground mod-cls). Since the hyperuniverse, the collection of all countable transitive modcls AB) of ZFC, is closed under all possible universe-creation methods, one is led to identifying the multiverse with it. Second, requiring in Desideratum I that

the multiverse be given a precise mathematical formulation enables one to put it to work for the aim of enriching the realm of set-thcoretic truth. This

is done in the Hyperuniverse Program by formulating justifiable preferences

for certain members of the hyperuniverse over others, thereby obtaining a

selection of preferred universes. The requirement that the multiverse bc

well-defined is a necessary condition for this selection process to be possible,

which would not be the case were the multiverse ill-defined or open-ended.

DESIDERATUM 2. The hyperuniverse is not an ultimate plurality. One can

express preferences for certain members of it according to criteria based on justified principles.

Another key point in the Hyperuniverse Program is that first-order prop-erties which are true across preferred universes of the hyperuniverse are truc

in V.

DESIDERATUM 3. Any first-order property of V is reflected into a countable transitive model of ZFC which is a preferred member of the hyperuniverse.

An important conscqucnce of Desideratum 3 is that, while the criteria

for preferred universes formulated within the Hyperuniverse Program may

be non first-order(indeed the criteria that we will introduce in Section 3

arc not—they quantify over the entire hyperuniverse), nonetheless in the

Hyperuniverse Program one arrives at first-order axioms for set theory, these

being the first-order truths shared by the preferred universcs.

In justifying Desideratum 3 one may invoke the downward Lowenheim-

Skolcm theorem, which, however, per se only implies that there must be

members of the hyperuniverse into which V first-order reflects. That these

may be chosen as preferred clements of the hyperuniverse is an assumption

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