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THE BuLLmN or SYMDouc Loac

Volame 19, Nam.ber 1,March2013

THE HYPERUNIVERSEPROGRAM

TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

Abstract. The Hyperaiwrse Program is a new approach to sel-theoretic truth which is based on justifiable principles and Jeads to the resolution of mamy questions independent from ZFC. The purpose of this paper is to present this program.to illustrate its machematical contept and implications,and to discuss its phitosophical assumptons.

§1. Introduction. The purpose of this paper is to discuss and illustrate the Hyperuniverse Program(as well as the Inner Model Hypothesis (IMH) and its variants as a proposal for realizing it), an approach due to the second author (sce[7])which is inspired by the search for solutions to qucstions known to be independent from the axiomatic system ZFC. ˡ

In recent years, different research programs, motivated by independence phenomenahave been formulated in set theory The stage for most of them has been set by Gödel's program for new axioms, announced in [9]at a time when the independence of the Continuum Hypothesis from ZFC could only be (correctly) conjcctured. [9], and its revised and extendedversion[10].have played a fundamental role in the debate concerningthe foundations of set theory. In defense of the views there expressed. Gödel invoked philosophical considerations on the nature of mathematics,anal- ysis of logico-mathematical concepts, and technical arguments of a purely mathematical character.Similar ingredients can be found in most of the subsequent proposals for overcoming independence results.

Gödel's program is worth a closer look. As a fundamental motivation for the program of extending ZFC through the addition of new axioms, the conviction is expressed in [9]that it is possible to give a final answer

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　Received January 31，2012.

Key words and phrases independence, large cardinals, multiverse, Platonism and Anti- platonism, set-theoretic truth.

　Both authors were supported by the Austrian Science Fund (FWF), Project P22430-N13.The first author was also supported by the European Science Foundation by an Exchange Grant within the framework of the ESF activity ‘New Frontiers of Infinity: Mathematical. Philosophical and Computational Prospecta".

ˡ By independent questions we mean sentences ф of the first-order language of ZFC such that ZFC can prove neither ф nor ~ф.

　　Ⓒ2013, Assaciation fer Symbeic Lege1079-8986/13/1901-0003/$1.00

DOL:10.2178/bul.1901030

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78 TATIANA ARRIGONI AND SY-DAWID FRIEDMAN

to the question of the cardinality of the continuum,despite its probable

independence from ZFC. This conviction explicitly rests on a Platonistic view

of mathematics,according to which set-theoretical concepts and theorems

describe some well determined reality, “in which Cantor's conjecture must be

cither true or false,and its undecidability from the axioms as known today

can only mean that these axioms do not contain a complete description of

this reality”,([9].p.181).

When it comes to discussing proposals for new axioms, the point is made

in [9] that the candidates for new axioms should be justified,displaying

conformity to motivating principles more evident and persuasive than the

candidates themselves. The concept of set is called upon for this purpose, AB) where the view is taken that a set is somethingobtainable from the integers (orsome other well-defined object) by iterated application of the operation “set

of” ([9]. p. 180). A special emphasis is put on the maximizing implications

of that concept, to the effect that axioms “stating the existence of still further

iterations of the operation set of",like “small” large cardinal hypotheses.

are regarded as fully legitimate candidates for new set-theoretic axioms.

[9].however,does not rule out the possibility that, beyond the concept of

set, there may be other motivations that succeed in indicating reasonable

strategics for extending ZFC. In fact it is conjectured that “there may exist

besides the ordinary axioms [...] other (hitherto unknown) axioms of set

theory which a more profound understanding of the concepts underlying

logic and mathematics would enable us to recognize as implied by these

concepts”([9]. p. 182). The suggestion is also made, in [10], that some

maximum property of the system of sets may be devised that is not directly

suggested by the concept of set, yet may work as a reasonable new axiom for

set theory ("[...] from an axiom in some sense opposite to this one [V=L]

the negation of Cantor's conjecture could perhaps be derived. I am thinking

of an axiom which [...] would state some maximum property of the system

of all sets […]”. [10]. p.478).

It is by invoking the criterion of success as contributing to a decision about

the truth of a candidate for an axiom for set theory that the way is opened in

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　² By “small large cardinals" are here meant large cardinals whose existence is compatible

with the axiom of constructibility V = L.Gödel pronounces on one of them.the axiom

stating the existence of an inaccessible cardinal,as follows:

The simplest of thesc "strong axioms of infinity" asscrts the cxistence of inac-

cessible numbers (and of oumbers inaccessiblc in the stronger sensc)＞ℵ₀. The

lutter axiom, roughly speaking.means nothing else but that the totality of sets

obtainable by exclusive use of the procedure of formation of sets expressed in the

other axioms forms again a set (and therefore a new basis for further applications

of these processes). ([9], p.182)

The axiom stating the existence of a measurable cardinal, as well as its incompatibility with the

axiom of constructibility, was known to Gödel as he wrote [10].However, Gödel apparently

did not consider this axiom as implied by the coacept of set (see[10], footaote 16).

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　THE HYPERLNIVERSE PROGRAM 79

[9] for bringing considerations of a purely mathematical character into the

discussion of proposals for new axioms. The success of an axiom is meant to

consist in its fruitfulness in consequences, its "shedding light upon a wholediscipline", and its yielding “powerful methods for solving given problems"

([9]. p.183). Mathematical results (facts “not known at Cantor's time") are

also invoked in the attempt to explain the prediction that Cantor's conjecture

will turn out to be wrong. Thus, the moral of [9] is that in formulating axiom

candidates for set theory, one not only is committed to the search for general

motivating principles that justify them, but one must also take into account

a corpus of already existing and accepted mathematical results, upon which

the new axiom(s) should shed light, or at the very least, not irreconcilably

contradict.

The approach that we present here shares many features, though not all. ofGödel's program for new axioms.Let us briefly illustrate it.The Hyperuni-

verse Program is an attempt to clarify which first-order set-theoretic state- AB) ments (beyond ZFC and its implications) are to be regarded as true in V. bycreating a context in which different pictures of the set-theoretic universe can

be compared. This context is the hyperuniverse, defined as the collection of all

countable transitive models of ZFC. The comparison of such models evokes

principles(principles of maximaliry and onniscience, as we will name two

of them) that suggest criteria for preferring. on justifiable grounds,certain

universes of sets over others.³ Starting from criteria for preferred universes,

one applies the principle that first-order statements that hold across all pre-

ferred universes (hopefully including solutions to independent questions)

also hold in V (an assumption based partly on the downward Lowenheim-

Skolem theorem), and adopts these statements as new axioms of set theory.

This being, in a nutshell, the Hyperuniverse Program, one clearly sees that

it shares the fundamental aim of Gödel's program of extending ZFC by newset-theoretic axioms resulting from "a more profound understanding of basic

concepts underlying logic and mathematics".In fact,within the Hyperuni-

verse Program one formulates principles and criteria for preferred universes

that are suggested by a logico-mathematical analysis of the hyperuniverse.

Also, Gödel's suggestion to consider a “maximn property of the system of

all sets" for extending ZFC is addressed by this program. Indeed maximality

works well as a principle inspiring criteria for preferred universes.Moreover,

in both Gödel's program and the Hyperuniverse Program, one seeks to find

solutions to independent questions in a way that may be regarded as ultimate

and not revisable, and hence may be regarded as definitive or true in V，the

universe of all sets.

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"The formulation of criteria for preferred universes is not an casy task. In particular the

possibility of conflicting desiderata to be imposed on preferred universes of sets cannot be

excluded at the outset.Essential to the Hyperuniverse Program is thus the effort to combine

the desired criteria into a coherent synthesis; this will be explained in detail below.

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80　　 TATIANA ARRIGONI AND SY-DAVID FRIEDMAN

It must be explicity said, however, that in formulating the Hyperuniverse

Program, Platonism is nowhere invoked, cither with regard to Vor to the

hyperuniverse. To the contrary, some of its characteristic features clearly

express an anti-Platonistic attitude, which makes the program radically dif-

ferent from Gödel's. No well-determined reality is called upon within the

Hyperuniverse Program in arguing for the legitimacy of the search for so-

lutions to independent questions. Rather, one considers that in spite of

the abundancc of independence results obtained in set theory, there are no

a priori grounds against the goal of finding ultimate answers to questions

like CH. This shifts the burden of proof onto those who claim that there are

such grounds.⁴ Moreover,in formulating the Hyperuniverse Program the

cxpression “trwe in V” is not used to reflcct an ontological state of affairs

concerning the universe of all sets as a reality to which existence can be as-

cribed independently of set-thcoretic practice.Instead “true in V” is meant

as a facon de parler that only conveys information about set-theorists'epis-

temic attitudes, as a description of the status that certain statements have or AB are expected to have in set-theorists'eyes. Sentences “true in V”are meant

to be sentences that are or should be regarded by set-theorists as definitive.

i.e,ultimate and not revisable.Within the Hyperuniverse Program two sorts

of statements qualify for this status. The first are those sct-thcorctic statc-

ments that, due to the role that they play in the practice of set theory and.

more gencrally, of mathematics, should not be contradicted by any further

candidate for a set-theoretic statement that may be regarded as ultimate and

not revisable. Let us call these statements "de facto”set-thcoretic truths.

The axioms of ZFC and the consistency of ZFC + large cardinal axioms are

examples of such truths.But secondly,within the Hyperuniverse Program,

one is ready to regard as true in statements that, beyond not contradicting

de facto set-theoretic truth, obey a condition for truth explicitly established

at the outset. Let us call these "de jure” set-theoretic truths. The condition

which they obey is that they are sentences that hold in all preferred universes

of the hyperuniverse. The latter, in turn, is not meant as an independent,

well-determined reality, but as a mathematical construct, produced along

with the developments of set theory and of the program. Hence, within the

Hyperuniverse Program,Platonism is invoked neither with regard to V nor

with regard to the hyperuniverse. In fact,as intended by the program,for-

mulating de jure set-thcoretic truths is an autonomouslyregulated process.

No“external” constraint is imposed while cngaged in it, such as an already

existing reality to which one must be faithful. Instead,in searching for de

jure set-theoretic truths one is only expected to follow justifiable procedures

It cannot be excluded at the outset that at some time the need will arisc to

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　This assumption scems to lie behind Shelah's considerations in [18]. [19]as well as the

mutiverse vlew advocated by Hamkins in [11]. A critical appraisal of the former is given

in [1].

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