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86 TATIANA ARRIGONI AND SYAVID FRIEDMAN

that is made ad hoc within the Hyperuniverse Program, arguing that it

expresses a reasonable procedure for cnlarging the realm of sct-theoretic

truth. There is no need in the Hyperuniverse Program to show that this strategy is the “right” one for arriving at ncw truths of set thcory. In fact no Platonistic assumption underlies the program, no commitment to a view of V as a well-determined reality existing independently of mathematical practicc

to which one should be faithful when extending set-theoretic knowledge

As a result, within the Hypcruniverse Program no a priori distinction is AB drawn between right and wrong strategies for arriving at new set-theoretic truths. Instead one aims to formulate and justify procedures for finding ncw set-theoretic statements that one wishes to regard as ultimate and definitive.

The reasonableness of the suggested procedure is the sole ground for the

claim that the statements arrived at deserve to be regarded as true in V.

Desiderata 2 and 3 amount to a proposal for a stratcgy toward finding

new set-theoretic truths (a proposal which,in its full form, has to include

explicit criteria for preferred clements of the hyperuniverse; we consider this

in Scction 3). How is one to argue for the reasonableness of this strategy?

Consider the aim of the Hyperuniverse Program. One wishes to master

the wide varicty of different pictures of I with which one is confronted in

contemporary sct theory,and which is faithfully represented by the hype-

runiverse. Due to the downward Lōwenheim-Skolem theorem, members

of the hyperuniverse are candidates for conveying first-orderinformation

about V.Being confronted with a bewildering number of different options

is a situation which we are familiar with not only in contemporary set theory.

A behavior which we naturally adopt in such a situation is the following:we

analyze what the possibilities are, choose among them those that under jus-tified criteria look better than others (hence could be privileged on a priori

grounds), and decide in favour of these. This is exactly what one does in the

Hyperuniverse Program. In one's scarch for new truths of IV one starts from

the hyperuniverse, which most faithfully reflects the possible pictures of the

set-thcoretic universe. As one is not content with the hyperuniverse as an

ultimate, non-transcendable context, one is led to the program described by

desiderata 2 and 3, which amounts to singling out members of the hyperuni-

verse that possess optimal meta-mathematical properties (i.e.,those which

obey the criteria for preferred universes), so as to decide in favour of them

for the purpose of enriching the realm of truth in V. The strategy of the

Hyperuniverse Program is thus perfectly reasonable in light of its aims.

Let us emphasize that there is per se no guarantee that the criteria which we

list below will lead to new axioms that both resolve independent questions

and are compatible with de facto set-theoretic truth. I.e., by following them

one is not sure at the outset that one will succeed in enlarging the realm

of truth in V beyond the sentences that are already accepted as definitive

in set theory. This is the result of the unbiased nature of the criteria for

preferred universes being used. However, it turns out that by selecting

THE HYPERUNIVERSE PROGRAM 87

universes according to our suggested criteria, one indeed obtains solutions to independent questions without conflicting with existing definitive truths of sct theory. That this de facto happens may be invoked as a relevant a posteriori argument (an argument from success) for the reasonableness of the strategy suggested by the Hyperuniversc Program.

§3. Criteria for preferred universes. Which universes are preferred in the Hyperuniverse Program?

In Section I we made the point that by subscribing to the Hyperuniverse Program one is expected to conform to principles and criteria for preferred universes that arise from an unbiased look at the hyperuniverse, so as to obtain a selection of universes that is justifiable. The program thus excludes the possibility that needs arising from specific areas of set-theoretic or math- cmatical practice play a role in formulating criteria for preferred universes. Therefore statements to the effect that one should prefer universes in which principles hold that resolve the difficulties arising in a specific area of set theory or mathematics are not candidates for such criteria. Let us give some examples of such non-criteria.

a. The generalised continuum hypothesis (GCH), which is very effective in resolving a wide range of questions in set theory; ¹¹

b.V=L.a theory which yields a powerful infinitary combinatorics which can be used to resolve even more problems in set theory than GCH;

c.Projcctive Determinacy(PD),which yiclds an attractive theory of pro- jective sets of reals;

d. Forcing axioms (such as MA, BPFA, BMM), which, like V = L.have great combinatorial strength. ¹²

Criteria of this kind reflect the interests of specific groups of set-thcorists or mathematicians, As a result, there may be as many different such criteria as there are areas of set theory or mathematics. Moreover, as interests in set thcory change, so may these criteria. Thus at the outsct, no sclection of universes can be made according to them that can presume to be universally recognized as lcgitimate within the set-thcoretic community as a whole. Is there a better way to select preferred universes?

The positive answer to this question given by the Hyperuniverse Program is that by focusing instead only on the most general features of the hype- runiverse and formulating principles based upon them,one is capable of suggesting (and justifying) criteria for preferred universes. This is based on the obvious fact that the hyperuniverse consists of ZFC models that may be

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¹¹See.e.g.[19) on the advantages of assuming GCH as an axiom.

¹²One may add to the list Woodin's axiomatic proposals and conjectures, introduced in

[22], based on Ω-logic. The latter is a logic which can be proved (under the assumption of the existence of a proper class of Woodin cardinals) to be unaffocted by sct-forcing. But as discussed earlier, one cannot justify an exclusive focus on set-forcing in suggesting new axioms and conjectures.

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88 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

mutually related (some universes may be,eg,forcing extensions, ground

models, or rank initial segments of others), and one may justifiably choose

elements of the hyperuniverse that are “prefcrable"in terms of this com-

parison. These are explicitly identified with the universes that, with respect

to those to which they are related, satisfy principles such as maximality or

omniscience.

Before considering how an element of the hyperuniverse may succeed in

being maximal.let us mention a danger of selecting universes according to

principles and criteria derived from an unbiascd look at the hyperuniversc.

In doing so one may be led to the adoption of first-order statements which contradictde facto set-thcoretic truth.Let us give an example One may

wish to make a selection of preferred universes based on a principle of min- AB) imaliry.One's criterion would therefore be that preferred universes should be as small as possible.This criterion may lead to the choice of just one uni-verse, the minimal model of ZFC, which would have as an implication that the statement that set models of ZFC do not exist expresses a property of V.This is however in obvious conflict with set-theoretic practice.i.e.,the exis-

tence of set models of ZFC does belong to the realm of de facto set-theoretic truth. The same applies to a weaker criterion inspired by a minimality prin-ciple, according to which one should prefer universes that satisfy the axiom of constructibility. V = L.Although the axiom of constructibility does allow for the existence of set models of ZFC (and more), it does not allow for the existence of inner models of ZFC with measurable cardinals. This

too stands in conflict with set-theoretic practice, i.e.,the existence of such

modcls belongs to the realm of de facto set-thcoretic truth (the point will be further discussed in the Appendix).

We turn now to the principle of maximality. A first point to make about

maximality is that one cannot have “structural maximality” within the hy-

peruniversc,in the sense that a preferred universe should contain all possible ordinals or real numbers. For there is no tallest countable transitive model of ZFC and over any such model new reals can be added to obtain another such model. What principle of maximaliry may be then imposed on elements of the hyperuniverse?

(Logical) Maximaliry:let be a variable that ranges over the elements of

the hyperuniverse. is (logically) maximalifall set-thcoretic statements with certain parameters which hold externally, i.e., in some universe containing ∪ as a "subuniverse",also hold internally, i.e..in some"subuniverse" of ∪.

Depending on what one takes as parameters and what onc takes for the

concept of “subuniverse”, different criteria for maximal universes arise from(and are justified in light of) this principle.Here are two examples.

• Criterion of ordinal (or vertical) maximaliry:this criterion appeals to

maximality with respect to the ordinals, where models have fixed the

power-set operation. Let us define a universe w to be a lengthening of

v ifv is a (proper) rank initial segment ofw. is ordinal maximal iff it

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THE HYPERUNIVERSE PROGRAM 89

has a lengthening w such that for all first-order formulas φ and subsets A of ∪ belonging to ω, if φ(A ∩ ∪α ) holds in ∪ᵦ then e(A n va) holds in ua for some pair of ordinals α ＜ β in ∪α (where ta denotes the collection of sets in e of rank less than α).

• Criterion of power set (or horizonial) maximaliry: this criterion appeals to maximality with respect to power set.where models have fixed ordi- nals. If a parameter-free sentence holds in some outer model of ∪ (i.e., in some universe w containing ∪ with the same ordinals as ∪), then it holds in some inner model of ∪ (i.e., in some universe to contained in ∪ with the same ordinals as ∪).

Ordinal (or vertical) maximality has a long history in set theory. It is also known as a higher-order reflection principle, and has been shown to imply (and to justify) the existence of “small”large cardinals (i.e.large cardi- nal notions consistent with V = L such as inaccessibles, weak compacts, w-Erdos cardinals, ...).¹³ Power set maximality, instead, has been only recently formulated. In fact it is equivalent to the IMH.which formally speaking states that by passing to an outermodelof ∪.internal consistency remains unchanged，i.e,the set of parameter-free sentences which hold in some inner model of ∪ is not increased. Assessing the compatibility of power set maximality with de facto set-theoretic truths is no trivial mat- ter. For the IMH refutes the existence of inaccessible cardinals as well as projective determinacy (PD) (see [7]).These implications have forced a re-examination of the roles of large cardinals and determinacy in set- thcoretic practice. As a result onc sees that power set maximality maybe compatible with de facto set-theoretic truths after all. For, if one accepts that the role of large cardinals in set theory is correctly described by saying that their existence in inner models, and not their existence in V, is a de facto set-theoretic truth, and that the importance of PD is captured by its parameter-free version, then the compatibility of power set maximality with set-theoretic practice is restored:the IMH is in fact consistent both with inner models of very large cardinals and with parameter-free PD (indeed with OD-determinacy without real paramcters).¹⁴ We will returm to the

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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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-

verses? We have so far formulated two candidate criteria: ordinal maxi-mality and power set maximality. The ideal situation would be to combine

them into a singlc consistent critcrion. i.e., a criterion that is satisficd by at

least one element of the hyperuniverse. This is not trivial, since power set

maximality and ordinal maximality contradict cach other. One is thus led

to the following conjecture:

SYNTHESIS CONJECTURE. Let power set maximality* (IMH*) be power set AB) maximality(IMH) restricted to ordinal maximal universes (ie.,the state-ment that if a sentence holds in an ordinal maximal outer model of ∪ then it holds in an inner model of ∪). Then the conjunction of power set maximal-ity* (IMH*) and ordinal maximality is consistent. l.e., there are universes which simultancously satisfy both criteria.

A proof of the Synthesis conjecture is within reach, as it only dcmands the

existing method for proving the consistency of the IMH (see [8])together

with a careful understanding of how Jensen coding can be done in the

presence of small large cardinal properties. Via the Hyperuniverse Program

the Synthesis Conjecture is effective in yielding new (first-order) set-theoretic

axioms, including solutions to independent questions. As universes which

witness the Synthesis Conjecture (i.e., which arc ordinal maximal and satisfy

IMH*) are preferred universes, first-order properties shared by all such

universes are true in I and may be adopted as new axioms. Examples of

such statements are the following (see [7].[8].[1]):

1. There are small large cardinals and inner models with measurable car-

dinals of arbitrary Mitchell order.

2. For some real R. R# does not exist and so Jensen covering holds with

respect to L[R],the constructible universe relativised to R.As a result:

3.There are no measurable cardinals, the singular cardinal hypothesis is

truc, the continuum is not real-valued measurable,projective deter-

minacy (PD) is false, the proper forcing axiom is false and there are

non-Borel analytic sets which are not Borel isomorphic.

The Contiman Hypothesis remains undecided, cven assuming that there

exists a universe that obeys the Synthesis Conjecture.One needs a stronger

version of power set maximality than the Inner Model Hypothesis to settle

CH. i.e,the hypothesis for formulas with globally absolute parameters.¹⁵ A

consistency proof for the resulting Strong Inner Model Hypothesis(SIMH)

is however still lacking.

Omniscience and a Grander Synthesis? Yet another source of criteria for

preferred universes is the principle of omniscience A universe is omniscient

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¹⁵See [8].

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