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THE HYTERUNIVERSE FROGRAM 91

if it is ablc to describe what can be true in alternative universes, A precisc

criterion based upon this principle is the following.

CRITERION OF OMNISCIENCE, Let Φ be the set of sentences with arbitrary

parameters from ∪ which can hold in some outer model of ∪ e. Then Φ is

first-order definable in ∪.

This kind of statement first appeared in unpublished work of Mack Stan-

ley, where he shows that there are omniscient universes (in our terminology). assuming a bit less than the consistency of a measurable cardinal (stationary-many Ramsey cardinals, roughly speaking). One may be tempted to regard omniscicnce as a form of power set maximality; this is unlikely, however, for whereas power-set maximality does not allow any parameters,the principle

of omniscience allows arbitrary set parameters.

Synthesizing ordinal maximality with omniscience should not be diflicult.

using indiscernibles for the Dodd-Jensen core model in the presence of Ramsey cardinals. An intriguing open question is how to achieve a grander synthesis with power set maximality. The obvious approach, asserting power set maximality for omniscient and ordinal maximal universes, appears to be

inconsistent. It is nevertheless reasonable to conjecture that some such grand

synthesis is possible. but its formulation will be subtle and the mathematics

required to verify consistency may be challenging.

§4. Conclusions. The Hyperuniverse Program presented in this paper is a

new approach to set-thcoretic truth, aimed at enlarging the realm oftrue-in- I

statements beyond ZFC. To this purpose the program develops a justifiable strategy, and regards the intrinsic reasonableness of this strategy as a guar-antee for the truth of the results obtained. More precisely, one introduces the hyperuniverse as the most suitable realization of the multiverseconcept and puts it to work for the purpose of comparing different pictures of the set-theoretic universe (countable transitive models of ZFC)in light of crite-ria for prefcrring some universes over others. First-order properties shared by all preferred universes are taken to be true in V.By invoking the criteria of ordinal (vertical) maximality and power sct (horizontal) maximality, a suitable realization of the program is obtained. By postulating the exis-tence of an element of the hyperuniversc that satisfies the natural synthesis of these criteria (i.e., the Synthesis Conjecture), one arrives at statements which are true in V yet independent from ZFC. These statements contradict the existence of very large cardinals but are consistent with their existence in inner models, and they contradict projective determinacy but are consistent

with determinacy for sets of reals which are ordinal-definable without real

parameters. This leads to a reassessment of the roles of large cardinals and determinacy in sct theory.

It is worth observing that, although the realization of the Hyperuni-verse Program presented in this paper fails to resolve many interesting

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TATIANA ARRICONI AND SY-DAVID FRIEDMAN 92

ZFC-independent questions and raises issues that call for further investi-gation (starting with the consistency of the Synthesis Conjecture),this by no means undermines the overall validity and mathematical fruitfulness of the program. Quite the contrary, the rescarch outcomes obtained and the questions entailed by the developments inspired by the Hyperuniverse Pro- gram attest to its mathematical potential and speak of its promise for the future,as further principles(such as omniscience) that motivate criteria for preferred universes are analyzed and discovered, and a synthesis is sought for them in conjunction with maximality.

§5. Appendix: the byperuniverse program, maximality, large cardinals and PD. This Appendix is devoted to a closer examination of the rela- tion of the Hyperuniverse Program to alternative proposals for extending set-thcoretic truth (beyond ZFC and other de facto true set-theoretic state- ments), in particular to large cardinals and Projective Determinacy (PD) as candidates for axioms of set theory.

Gödel's Program for new axioms, sketched in Section 1, includes the rec-

ommendation to consider some maximum property of the system of all sets for the purpose of extending ZFC. Since maximality is used in the Hyperuni-versc Program as a motivating principle for criteria for preferred universes we advocated above that this program meets Gödel's recommendation.Of

coursc, while claiming this,we are aware of the fact that the considerations

concerning maximality developed within the Hyperuniverse Program are

of a different nature than those invoked in alternative proposals for new

set-theoretic axioms. This applies, in particular to proposals to the efTect

that ZFC should be cnlarged through the addition of suitable large cardinal

hypotheses,as these being faithful to our expectations concerning the mix/-

mim character of the universe of all sets. Consider the following quotition

by H.Wang ([21].p.553):

We believe that the collection of all ordinals is very ‘long' and cach

power set (of an infinite) set is very ‘thick'. Hence any axioms to

such effect arc in accordance with our intuitive concept.

By giving, as Wang does, the length of the ordinals and the thickness of

powersets as examplesof maximumproperties of the systenofallsets.

oneipso factostarts from the assumptionthatby“maximum properties of the system of all sets" one means ontological features of Y related to what“exists”within it. In making this assumption, one may cither intend P as an independently existing well-determined reality (this seems to be the choice of Godcl in [9]), or (at least partly) as a well-determined epistemic notion,a mental representation of the universe which we are naturally led to by our intuitions concerning sets (Wang seems to see V in [21] in this way appealing to the iterative concept ofset). In cither case one can only realize the idea that the system of sets displays maximman properties by making

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