aa r X i v : . [ m a t h . L O ] O c t July 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
A multiverse perspective on the axiom of constructiblity
Joel David Hamkins a Visiting Professor of Philosophy, New York UniversityProfessor of Mathematics, The City University of New YorkThe Graduate Center & College of Staten [email protected], http://jdh.hamkins.org
I shall argue that the commonly held V = L via maximize position,which rejects the axiom of constructibility V = L on the basis thatit is restrictive, implicitly takes a stand in the pluralist debate in thephilosophy of set theory by presuming an absolute background conceptof ordinal. The argument appears to lose its force, in contrast, on anupwardly extensible concept of set, in light of the various facts showingthat models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.
1. Introduction
Set theorists often argue against the axiom of constructibility V = L on thebasis that it is restrictive. Some argue that we have no reason to think thatevery set should be constructible, or as Shelah puts it, “Why the hell shouldit be true?” [19]. To suppose that every set is constructible is seen as anartificial limitation on set-theoretic possibility, and perhaps it is a mistakenprinciple generally to suppose that all structure is definable. Furthermore, a This article expands on an argument that I made during my talk at the AsianInitiative for Infinity: Workshop on Infinity and Truth, held July 25–29, 2011 atthe Institute for Mathematical Sciences, National University of Singapore. Thiswork was undertaken during my subsequent visit at NYU in Summer and Fall,2011, and completed when I returned to CUNY. My research has been sup-ported in part by NSF grant DMS-0800762, PSC-CUNY grant 64732-00-42 and Si-mons Foundation grant 209252. Commentary concerning this paper can be made athttp://jdh.hamkins.org/multiverse-perspective-on-constructibility.1uly 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L2
J. D. Hamkins although V = L settles many set-theoretic questions, it seems so often tosettle them in the ‘wrong’ way, without the elegant smoothness and uni-fying vision of competing theories, such as the situation of descriptive settheory under V = L in comparison with that under projective determinacy.As a result, the constructible universe becomes a pathological land of coun-terexamples. That is bad news, but it could be overlooked, in my opinion,were it not for the much worse related news that V = L is inconsistentwith all the strongest large cardinal axioms. The boundary between thoselarge cardinals that can exist in L and those that cannot is the thresholdof set-theoretic strength, the entryway to the upper realm of infinity. Sincethe V = L hypothesis is inconsistent with the largest large cardinals, itblocks access to that realm, and this is perceived as intolerably limiting.This incompatibility, I believe, rather than any issue of definabilism or de-scriptive set-theoretic consequentialism, is the source of the most stridentend-of-the-line deal-breaking objections to the axiom of constructibility. Settheorists simply cannot accept an axiom that prevents access to their bestand strongest theories, the large cardinal hypotheses, which encapsulatetheir dreams of what our set theory can achieve and express.Maddy [14 ,
15] articulates the grounds that mathematicians often use inreaching this conclusion, mentioning especially the maximize maxim, saying“the view that V = L contradicts maximize is widespread,” citing Drake,Moschovakis and Scott. Steel argues that “ V = L is restrictive, in thatadopting it limits the interpretative power of our language.” He points outthat the large cardinal set theorist can still understand the V = L believerby means of the translation ϕ ϕ L , but “there is no translation in theother direction” and that “adding V = L . . . just prevents us from asking asmany questions!” [20]. At bottom, the axiom of constructibility appears tobe incompatible with strength in our set theory, and since we would like tostudy this strength, we reject the axiom.Let me refer to this general line of reasoning as the V = L via maxi-mize argument. The thesis of this article is that the V = L via maximizeargument relies on a singularist as opposed to pluralist stand on the ques-tion whether there is an absolute background concept of ordinal, that is,whether the ordinals can be viewed as forming a unique completed totality.The argument, therefore, implicitly takes sides in the universe versus mul-tiverse debate, and I shall argue that without that stand, the V = L viamaximize argument lacks force.In [16], Maddy gives the V = L via maximize argument sturdier legs,fleshing out a more detailed mathematical account of it, based on a method- uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L ology of mathematical naturalism and using the idea that maximizationinvolves realizing more isomorphism types. She begins with the ‘crude ver-sion’ of the argument:The idea is simply this: there are things like 0 ♯ that are not in L.And not only is 0 ♯ not in L; its existence implies the existence ofan isomorphism type that is not realized by anything in L.. . . So itseems that ZFC + V = L is restrictive because it rules out the extraisomorphism types available from ZFC + ∃ ♯ . [16]For the full-blown argument, she introduces the concept of a ‘fair interpre-tation’ of one theory in another and the idea of one theory maximizing overanother, leading eventually to a proposal of what it means for a theory tobe ‘restrictive’ (see the details in section 2), showing that ZFC + V = L andother theories are restrictive, as expected, in that sense.My thesis in this article is that the general line of the V = L via maxi-mize argument presumes that we have an absolute background concept ofordinal, that the ordinals build up to form an absolute completed totality.Of course, many set-theorists do take that stand, particularly set theoristsin the California school. The view that the ordinals form an absolute com-pleted totality follows, of course, from the closely related view that there isa unique absolute background concept of set, by which the sets accumulateto form the entire set-theoretic universe V , in which every set-theoretic as-sertion has a definitive final truth value. Martin essentially argues for theequivalence of these two commitments in his categoricity argument [17],where he argues for the uniqueness of the set-theoretic universe, an argu-ment that is a modern-day version of Zermelo’s categoricity argument withstrong parallels in Isaacson’s [11]. Martin’s argument is founded on theidea of an absolute unending well-ordered sequence of set-formation stages,an ‘Absolute Infinity’ as with Cantor. Although Martin admits that ‘it isof course possible to have doubts about the sharpness of the concept ofwellordering,” [17] , his argument presumes that the concept is sharp, justas I claim the V = L via maximize argument does.Let me briefly summarize the position I am defending in this article,which I shall describe more fully section in 4. On the upwardly extensibleconcept of set, one holds that any given concept of set or set-theoretic uni-verse may always be extended to a much better one, with more sets andlarger ordinals. Perhaps the original universe even becomes a mere count-able set in the extended universe. The ‘class of all ordinals’, on this view,makes sense only relative to a particular set-theoretic universe, for there is uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L4
J. D. Hamkins no expectation that these extensions cohere or converge. This multiverseperspective resonates with or even follows from a higher-order version ofthe maximize principle, where we maximize not merely which sets exist,but also which set-theoretic universes exist. Specifically, it would be lim-iting for one set-theoretic universe to have all the ordinals, when we canimagine another universe looking upon it as countable. Maximize therebyleads us to expect that every set-theoretic universe should not only haveextensions, but extremely rich extensions, satisfying extremely strong the-ories, with a full range of large cardinals. Meanwhile, I shall argue, themathematical results of section 3 lead naturally to the additional conclu-sion that every set-theoretic universe should also have extensions satisfying V = L . In particular, even if we have very strong large cardinal axioms inour current set-theoretic universe V , there is a much larger universe V + inwhich the former universe V is a countable transitive set and the axiomof constructibility holds. This perspective, by accommodating both largecardinals and V = L in the multiverse, appears to dissolve the principalthrust of the V = L via maximize argument. The idea that V = L is per-manently incompatible with large cardinals evaporates when we can havelarge cardinals and reattain V = L in a larger domain. In this way, V = L no longer seems restrictive, and the upward extensible concept of set revealshow large cardinals and other strong theories, as well as V = L , may all bepervasive as one moves up in the multiverse.
2. Some new problems with Maddy’s proposal
Although my main argument is concerned only with the general line of the V = L via maximize position, rather than with Maddy’s much more specificaccount of it in [16], before continuing with my main agument I wouldnevertheless like to mention a few problems with that specific proposal.To quickly summarize the details, she defines that a theory T shows ϕ isan inner model if T proves that ϕ defines a transitive class satisfying everyinstance of an axiom of ZFC , and either T proves every ordinal is in theclass, or T proves that there is an inaccessible cardinal κ , such that everyordinal less than κ is in the class. Next, ϕ is a fair interpretation of T in T ′ ,where T extends ZFC , if T ′ shows ϕ is an inner model and T ′ proves everyaxiom of T for this inner model. A theory T ′ maximizes over T , if thereis a fair interpretation ϕ of T in T ′ , and T ′ proves that this inner modelis not everything (let’s assume T ′ includes ZFC ). The theory T ′ properlymaximizes over T if it maximizes over T , but not conversely. The theory uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L T ′ strongly maximizes over T if the theories contradict one another, T ′ maximizes over T and no consistent extension T ′′ of T properly maximizesover T ′ . All of this culminates in her final proposal, which is to say thata theory T is restrictive if and only if there is a consistent theory T ′ thatstrongly maximizes over it.Let me begin with a quibble concerning the syntactic form of her def-inition of ‘shows ϕ is an inner model’, which in effect requires T to settlethe question of whether the inner model is to contain all ordinals or insteadmerely all ordinals up to an inaccessible cardinal. That is, she requires thateither T proves that ϕ is in the first case or that T proves that ϕ is in thesecond case, rather than the weaker requirement that T prove merely that ϕ is in one of the two cases (so the distinction is between ( T ⊢ A ) ∨ ( T ⊢ B )and T ⊢ A ∨ B ). To illustrate how this distinction plays out in her proposal,consider the theory Inacc = ZFC +‘there are unboundedly many inaccessi-ble cardinals’ and the theory T = ZFC +‘either there is a Mahlo cardinal orthere are unboundedly many inaccessible cardinals in L .’ (I shall assumewithout further remark that these large cardinal theories and the others Imention are consistent.) Every model of T has an inner model of Inacc , ei-ther by truncating at the Mahlo cardinal, if there is one, or by going to L , ifthere isn’t. Thus, we seem to have inner models of the form Maddy desires.Unfortunately, however, this is not good enough, and I claim that Inacc isactually not fairly interpreted in T . To see this, notice first that T does notprove the existence of an inaccessible cardinal, since we can force over anymodel of Inacc by destroying all inaccessible cardinals and thereby producea model of T having no inaccessible cardinals. b Consequently, if T shows ϕ is an inner model, it cannot be because of the second clause, which requires T to prove the existence of an inaccessible cardinal. Thus, T must prove ϕ holds of all ordinals. But notice also that T does not prove that there areunboundedly many inaccessible cardinals in L , since by truncation we caneasily have a Mahlo cardinal in L with no inaccessible cardinals above it.So T also cannot prove that ϕ defines a proper class model of Inacc . Thus,
Inacc is not fairly interpreted in T , even though we might have wished itto be. This issue can be addressed, of course, by modifying the definitionof shows-an-inner-model to subsume the disjunction under the provabilitysign, that is, by requiring instead that T prove the disjunction that either b First force ‘Ord is not Mahlo’ by adding a closed unbounded class C of non-inaccessiblecardinals—this forcing adds no new sets—and then perform Easton forcing to ensure2 γ = δ + whenever γ is regular and δ is the next element of C .uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L6
J. D. Hamkins ϕ holds of all ordinals or that it holds of all ordinals up to an inaccessiblecardinal. But let me leave this issue; it does not affect my later comments.My next objection is that the fairly-interpreted-in relation is not tran-sitive, whereas our pre-reflective ideas for an interpreted-in relation wouldcall for it to be transitive. That is, I claim that it can happen that a firsttheory has a fair interpretation in a second, which has a fair interpretationin a third, but the first theory has no fair interpretation in the third. Hereis a specific example showing the lack of transitivity: R = ZFC + V = L + there is no inaccessible cardinal S = ZFC + V = L + there is an inaccessible cardinal T = ZFC + ω is inaccessible in L The reader may easily verify that R has a fair interpretation in S by trun-cating the universe at the first inaccessible cardinal, and S has a fair inter-pretation in T by going to L . Furthermore, every model of S has forcingextensions satisfying T , by the L´evy collapse. Meanwhile, I claim that R has no fair interpretation in T . The reason is that T is consistent with thelack of inaccessible cardinals, and so if T shows ϕ is an inner model, thenin any model of T having no inaccessible cardinals, this inner model mustcontain all the ordinals. In this case, in order for it to have R ϕ , the innermodel must be all of L , which according to T has an inaccessible cardinal,and therefore doesn’t satisfy R after all. So R is not fairly interpreted in T . The reader may construct many similar examples of intransitivity. Theessence here is that the first theory is fairly interpreted in the second onlyby truncating, and the second is fairly interpreted in the third only by go-ing to an inner model containing all the ordinals, but there is no way tointerpret the first in the third except by doing both, which is not allowed inthe definition if the truncation point is inaccessible only in the inner modeland not in the larger universe.The same example shows that the maximizing-over relation also is nottransitive, since T maximizes over S and S maximizes over R , by the fairinterpretations mentioned above (note that these theories are mutually ex-clusive), but T does not maximize over R , since R has no fair interpreta-tion in T . Similarly, the reader may verify that the example shows that theproperly-maximizes-over and the strongly-maximizes-over relations also arenot transitive.Let me turn now to give a few additional examples of what Maddy callsa ‘false positive,’ a theory deemed formally restrictive, which we do not findintuitively to be restrictive. As I see it, the main purpose of [16] is to give uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L precise mathematical substance to the intuitive idea that some set theoriesseem restrictive in a way that others do not. We view V = L and ‘thereis a largest inaccessible cardinal’ as limiting, while ‘there are unboundedlymany inaccessible cardinals’ seems open-ended and unrestrictive. Maddypresents some false positives, including an example of Steel’s showing that ZFC +‘there is a measurable cardinal’ is restrictive because it is stronglymaximized by the theory
ZFC + 0 † exists + ∀ α < ω L α [0 † ] = ZFC . L¨owepoints out that “this example can be generalized to at least every inter-esting theory in large cardinal form extending
ZFC . Thus, most theoriesare restrictive in a formal sense,” [12] and he shows in [13] that
ZFC it-self is formally restrictive because it is maximized by the theory ZF +‘everyuncountable cardinal is singular’.I would like to present examples of a different type, which involve whatI believe to be more attractive maximizing theories that seem to avoidthe counterarguments that have been made to the previous examples offalse positives. First, consider again the theory Inacc , asserting
ZFC +‘thereare unboundedly many inaccessible cardinals’, a theory Maddy wants toregard as not restrictive. Let T be the theory asserting ZFC +‘there areunboundedly many inaccessible cardinals in L , but no worldly cardinals in V .’ A cardinal κ is worldly when V κ | = ZFC . Worldliness is a weakening ofinaccessibility, since every inaccessible cardinal is worldly and in fact a limitof worldly cardinals; but meanwhile, worldly cardinals need not be regular,and the regular worldly cardinals are exactly the inaccessible cardinals.The worldly cardinals often serve as a substitute for inaccessible cardinals,allowing one to weaken the large cardinal commitment of a hypothesis. Forexample, one may carry out most uses of the Grothendieck universe axiomin category theory by using mere worldly cardinals in place of inaccessiblecardinals. The theory T is equiconsistent with Inacc , since every modelof
Inacc has a class forcing extension of T . c The theory
Inacc has a fairinterpretation in T , by going to L , and as a result, T maximizes over Inacc .Meanwhile, I claim that no strengthening of
Inacc properly maximizes over T . To see this, suppose that Inacc + contains Inacc and shows ϕ is an innermodel M satisfying T . If M contains all the ordinals, then since Inacc proves that the inaccessible cardinals are unbounded, M would have tocontain all those inaccessible cardinals, which would remain inaccessible c One first adds a closed unbounded class C of cardinals containing no worldly cardinals(this forcing adds no new sets), and then performs Easton forcing so as to ensure that2 γ = δ + , where γ is regular and δ is the next element of C . The result is a model of T ,since all worldly cardinals have been killed off.uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L8
J. D. Hamkins in M since inaccessibility is downward absolute, and therefore violate theclaim of T that there are no worldly cardinals. So by the definition of fairinterpretation, therefore, M would have to contain all the ordinals up toan inaccessible cardinal κ . But in this case, a Loweheim-Skolem argumentshows that there is a closed unbounded set of γ < κ with V Mγ ≺ V Mκ , andall such γ would be worldly cardinals in M , violating T . Thus, Inacc isstrongly maximized by T , and so Inacc is restrictive.Let me improve the example to make it more attractive, provided thatwe read Maddy’s definition of ‘fair interpretation’ in a way that I believeshe may have intended. The issue is that although Maddy refers to ‘trun-cation. . . at inaccessible levels’ and her definition is typically described byothers using that phrase, nevertheless the particular way that she wrote herdefinition does not actually ensure that the truncation occurs at an inac-cessible level. Specifically, in the truncation case, she writes that T shouldprove that there is an inaccessible cardinal κ for which ∀ α ( α < κ → ϕ ( α )).But should this implication be a biconditional? Otherwise, of course, noth-ing prevents ϕ from continuing past κ , and the definition would be moreaccurately described as ‘truncation at, or somewhere above, an inaccessiblecardinal’. If one wants to allow truncation at non-inaccessible cardinals,why should we bother to insist that the height should exceed some inac-cessible cardinal? Replacing this implication with a biconditional wouldindeed ensure that when the inner model arises by truncation, it does so bytruncating at an inaccessible cardinal level. So let us modify the reading of‘fair interpretation’ so that truncation, if it occurs, does so at an inaccessi-ble cardinal level. In this case, consider the theory Inacc as before, and let MC ∗ be the theory ZFC +‘there is a measurable cardinal with no worldlycardinals above it’. By truncating at a measurable cardinal, we produce amodel of
Inacc , and so MC ∗ offers a fair interpretation of Inacc , and con-sequently MC ∗ maximizes over Inacc . But no consistent strengthening of
Inacc can maximize over MC ∗ , since if V | = Inacc and W is an inner modelof V satisfying MC ∗ , then W cannot contain all the ordinals of V , since theinaccessible cardinals would be worldly in W , and neither can the heightof W be inaccessible in V , since if κ = W ∩ Ord is inaccessible in V , thenby a Lowenheim-Skolem argument there must be a closed unbounded set of γ < κ such that W γ ≺ W , and this will cause unboundedly many worldlycardinals in W , contrary to MC ∗ . Thus, on the modified definition of fairinterpretation, we conclude that MC ∗ strongly maximizes over Inacc , andso
Inacc is restricted.One may construct similar examples using the theory
ZFC +‘there is uly 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L a proper class of measurable cardinals’, which is strongly maximized by SC ∗ = ZFC +‘there is a supercompact cardinal with no worldly cardinalsabove it’. Truncating at the supercompact cardinal produces a model ofthe former theory, but no strengthening of the former theory can show SC ∗ in an inner model, since the unboundedly many measurable cardinals of theformer theory prevent the showing of any proper class model of SC ∗ , andthe eventual lack of worldly cardinals in SC ∗ prevents it from being shownin any truncation at an inaccessible level of any model of ZFC . A generalformat for these examples would be
ZFC +‘there is a proper class of largecardinals of type LC ’ and T = ZFC +‘there is an inaccessible limit of LC cardinals, with no worldly cardinals above.’ Such examples work for anylarge cardinal notion LC that implies worldliness, is absolute to truncationsat inaccessible levels and is consistent with a lack of worldly cardinals above.Almost all (but not all) of the standard large cardinal notions have thesefeatures.Maddy has rejected some of the false positives on the grounds thatthe strongly maximizing theory involved is a ‘dud’ theory, such as ZFC + ¬ Con(
ZFC ). Are the theories above, MC ∗ and SC ∗ , duds in this sense?It seems hard to argue that they are. For various reasons, set theoristsoften consider models of set theory with largest instances of large cardinalsand no large cardinals above, often obtaining such models by truncation,in order to facilitate certain constructions. Indeed, the idea of truncatingthe universe at an inaccessible cardinal level lies at the heart of Maddy’sdefinitions. But much of the value of that idea is already obtained when onetruncates at the worldly cardinals instead. The theory MC ∗ can be obtainedfrom any model of measurable cardinal by truncating at the least worldlycardinal above it, if there is one. But moreover, one needn’t truncate atall: one can force MC ∗ over any model with a measurable cardinal, by verymild forcing. First, add a closed unbounded class C of cardinals containingno worldly cardinal, and then perform Easton forcing so as to ensure inthe forcing extension that 2 γ = δ + , whenever γ is regular and δ is the nextelement of C above γ . The point is that this forcing will ensure that thecontinuum function γ γ jumps over the former worldly cardinals, and sothey will no longer be worldly (and no new worldly cardinals are created).If one starts this forcing above a measurable cardinal κ , then one preservesthat measurable cardinal while killing all the worldly cardinals above it.(In the case of SC ∗ , one should first make the supercompact cardinal Laverindestructible.) Because we can obtain MC ∗ and SC ∗ by moving from alarge cardinal model to a forcing extension, where all the previous context uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L10
J. D. Hamkins and strength seems still available, these theories do not seem to be duds inany obvious way. Nevertheless, the theories MC ∗ and SC ∗ are restrictive,of course, in the intuitive sense that Maddy’s project is concerned with.But to object that these theories are duds on the grounds that they arerestrictive would be to give up the entire project; the point was to giveprecise substance to our notion of ‘restrictive’, and it would beg the questionto define that a theory is restrictive if it is strongly maximized by a theorythat is not ‘restrictive.’
3. Several ways in which V=L is compatible with strength
In order to support my main thesis, I would like next to survey a series ofmathematical results, most of them a part of set-theoretic folklore, whichreveal various senses in which the axiom of constructibility V = L is com-patible with strength in set theory, particularly if one has in mind thepossibility of moving from one universe of set theory to a much larger one.First, there is the easy observation, expressed in observation 3.1, that L and V satisfy the same consistency assertions. For any constructibletheory T in any language—and by a ‘constructible’ theory I mean just that T ∈ L , which is true of any c.e. theory, such as ZFC plus any of the usuallarge cardinal hypotheses—the constructible universe L and V agree onthe consistency of T because they have exactly the same proofs from T . Itfollows from this, by the completeness theorem, that they also have modelsof exactly the same constructible theories.Observation 3.1: The constructible universe L and V agree on the con-sistency of any constructible theory. They have models of the same con-structible theories.What this easy fact shows, therefore, is that while asserting V = L we may continue to make all the same consistency assertions, such asCon( ZFC + ∃ measurable cardinal), with exactly the same confidence thatwe might hope to do so in V , and we correspondingly find models of ourfavorite strong theories inside L . Perhaps a skeptic worries that those mod-els in L are somehow defective? Perhaps we find only ill-founded models ofour strong theory in L ? Not at all, in light of the following theorem, a factthat I found eye-opening when I first came to know it years ago. Theorem 3.2:
The constructible universe L and V have transitive modelsof exactly the same constructible theories in the language of set theory. uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L Proof:
The assertion that a given theory T has a transitive model hascomplexity Σ ( T ), in the form “there is a real coding a well founded struc-ture satisfying T ,” and so it is absolute between L and V by the Shoenfieldabsoluteness theorem, provided the theory itself is in L .Consequently, one can have transitive models of extremely strong largecardinal theories without ever leaving L . For example, if there is a transitivemodel of the theory ZFC +“there is a proper class of Woodin cardinals,” thenthere is such a transitive model inside L . The theorem has the followinginteresting consequence. Corollary 3.3: (Levy-Shoenfield absoluteness theorem) In particular, L and V satisfy the same Σ sentences, with parameters hereditarily countablein L . Indeed, L ω L and V satisfy the same such sentences. Proof:
Since L is a transitive class, it follows that L is a ∆ -elementarysubstructure of V , and so Σ truth easily goes upward from L to V . Con-versely, suppose V satisfies ∃ x ϕ ( x, z ), where ϕ is ∆ and z is hereditarilycountable in L . Thus, V has a transitive model of the theory ∃ xϕ ( x, z ),together with the atomic diagram of the transitive closure z and a bijectionof it to ω . By observation 3.1, it follows that L has such a model as well.But a transitive model of this theory in L implies that there really is an x ∈ L with ϕ ( x, z ), as desired. Since the witness is countable in L , we findthe witness in L ω L .One may conversely supply a direct proof of corollary 3.3 via the Shoen-field absoluteness theorem and then view theorem 3.2 as the consequence,because the assertion that there is a transitive model of a given theory in L is Σ assertion about that theory.I should like now to go further. Not only do L and V have transitivemodels of the same strong theories, but what is more, any given model ofset theory can, in principle, be continued to a model of V = L . Considerfirst the case of a countable transitive model h M, ∈i . Theorem 3.4:
Every countable transitive set is a countable transitive set uly 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L12
J. D. Hamkins in the well-founded part of an ω -model of V = L . • M ML
Proof:
The statement is true inside L , since every countable transitive setof L is an element of some countable L α , which is transitive and satisfies V = L . Further, the complexity of the assertion is Π , since it assertsthat for every countable transitive set, there is another countable objectsatisfying a certain arithmetic property in relation to it. Consequently, bythe Shoenfield absoluteness theorem, the statement is true.Thus, every countable transitive set has an end-extension to a model of V = L in which it is a set. In particular, if we have a countable transitivemodel h M, ∈i | = ZFC , and perhaps this is a model of some very stronglarge cardinal theory, such as a proper class of supercompact cardinals,then nevertheless there is a model h N, ∈ N i | = V = L which has M as anelement, in such a way that the membership relation of ∈ N agrees with ∈ on the members of M . This implies that the ordinals of N are well-foundedat least to the height of M , and so not only is N an ω -model, but it is an ξ -model where ξ = Ord M , and we may assume that the membership relation ∈ N of N is the standard relation ∈ for sets of rank up to and far exceeding ξ . Furthermore, we may additionally arrange that the model satisfies ZFC − ,or any desired finite fragment of ZFC , since this additional requirement isachievable in L and the assertion that it is met still has complexity Π .If there are arbitrarily large λ < ω L with L λ | = ZFC , a hypothesis thatfollows from the existence of a single inaccessible cardinal (or merely froman uncountable transitive model of ZF ), then one can similarly obtain ZFC in the desired end-extension.A model of set theory is pointwise definable if every object in the modelis definable there without parameters. This implies V = HOD, since in factno ordinal parameters are required, and one should view it as an extremelystrong form of V = HOD, although the pointwise definability property,since it implies that the model is countable, is not first-order expressible.The main theorem of [10] is that every countable model of ZFC (and simi- uly 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L larly for GBC ) has a class forcing extension that is pointwise definable.
Theorem 3.5:
If there are arbitrarily large λ < ω L with L λ | = ZFC , thenevery countable transitive set M is a countable transitive set inside a struc-ture M + that is a pointwise-definable model of ZFC + V = L , and M + is wellfounded as high in the countable ordinals as desired. Proof:
See [10] for further details. First, note that every real z in L is in apointwise definable L α , since otherwise, the L -least counterexample z wouldbe definable in L ω and hence in the the Skolem hull of ∅ in L ω , whichcollapses to a pointwise definable L α in which z is definable, a contradiction.For any such α , let L λ | = ZFC have exactly α many smaller L β satisfying ZFC , and so α and hence also z is definable in L λ , whose Skolem hull of ∅ therefore collapses to a pointwise definable model of ZFC + V = L containing z . So the conclusion of the theorem is true in L . Since the complexity of thisassertion is Π , it is therefore absolute to V by the Shoenfield absolutenesstheorem.Theorems 3.4 and 3.5 admit of some striking examples. Suppose forinstance that 0 ♯ exists. Considering it as a real, the argument shows that0 ♯ exists inside a pointwise definable model of ZFC + V = L , well-foundedfar beyond ω L . So we achieve the bizarre situation in which the true 0 ♯ sitsunrecognized, yet definable, inside a model of V = L which is well-foundeda long way. For a second example, consider a forcing extension V [ g ] by theforcing to collapse ω to ω . The generic filter g is coded by a real, and so in V [ g ] there is a model M | = ZFC + V = L with g ∈ M and M well-foundedbeyond ω V . The model M believes that the generic object g is actuallyconstructible, constructed at some (necessarily nonstandard) stage beyond ω V . Surely these models are unusual.The theme of these arguments goes back, of course, to an elegant the-orem of Barwise, theorem 3.6, asserting that every countable model of ZF has an end-extension to a model of ZFC + V = L . In Barwise’s theorem, theoriginal model becomes merely a subset of the end-extension, rather than anelement of the end-extension as in theorems 3.4 and 3.5. By giving up on thegoal of making the original universe itself a set in the end-extension, Barwiseseeks only to make the elements of the original universe constructible in theextension, and is thereby able to achieve the full theory of ZFC + V = L inthe end-extension, without the extra hypothesis as in theorem 3.5, whichcannot be omitted there. Another important difference is that Barwise’s uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L14
J. D. Hamkins theorem 3.6 also applies to nonstandard models.
Theorem 3.6: (Barwise [2] ) Every countable model of ZF has an end-extension to a model of ZFC + V = L . M L
Let me briefly outline a proof in the case of a countable transitive model M | = ZF . For such an M , let T be the theory ZFC plus the infinitaryassertions σ a = ∀ z ( z ∈ ˇ a ⇐⇒ W b ∈ a z = ˇ b ), for every a ∈ M , in the L ω ,ω language of set theory with constant symbol ˇ a for every element a ∈ M .The σ a assertions, which are expressible in L ∞ ,ω logic in the sense of M ,ensure that the models of T are precisely (up to isomorphism) the end-extensions of M satisfying ZFC . What we seek, therefore, is a model of thetheory T + V = L . Suppose toward contradiction that there is none. I claimconsequently that there is a proof of a contradiction from T + V = L in theinfinitary deduction system for L ∞ ,ω logic, with such infinitary rules as:from σ i for i ∈ I , deduce V i σ i . Furthermore, I claim that there is such aproof inside M . Suppose not. Then M thinks that the theory T + V = L isconsistent in L ∞ ,ω logic. We may therefore carry out a Henkin constructionover M by building a new theory T + ⊆ M extending T + V = L , withinfinitely many new constant symbols, adding one new sentence at a time,each involving only finitely many of the new constants, in such a way soas to ensure that (i) the extension at each stage remains M -consistent; (ii) T + eventually includes any given L ∞ ,ω sentence in M or its negation, forsentences involving only finitely many of the new constants; (iii) T + hasthe Henkin property in that it contains ∃ x ϕ ( x, ~c ) = ⇒ ϕ ( d, ~c ), where d isa new constant symbol used expressly for this formula; and (iv) whenever adisjunct W i σ i is in T + , then also some particular σ i is in T + . We may buildsuch a T + in ω many steps just as in the classical Henkin construction. If N is the Henkin model derived from T + , then an inductive argument showsthat N satisfies every sentence in T + , and in particular, it is a model of T + V = L , which contradicts our assumption that this theory had no model.So there must be a proof of a contradiction from T + V = L in the deductivesystem for L ∞ ,ω logic inside M . Since the assertion that there is such aproof is Σ assertion in the language of set theory, it follows by the Levy- uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L Shoenfield theorem (corollary 3.3) that there is such a proof inside L M ,and indeed, inside L Mω . This proof is a countable object in L M and usesthe axioms σ a only for a ∈ L Mω . But L M satisfies the theory T + V = L and also σ a for all such a and hence is a model of the theory from whichwe had derived a contradiction. This violates soundness for the deductionsystem, and so T + V = L has a model after all. Consequently, M has anend-extension satisfying ZFC + V = L , as desired, and this completes theproof.We may attain a stronger theorem, where every a ∈ M becomes count-able in the end-extension model, simply by adding the assertions ‘ˇ a iscountable ’ to the theory T . The point is that ultimately the proof of acontradiction exists inside L Mω , and so the model L M satisfies these addi-tional assertions for the relevant a . Similarly, we may also arrange that theend-extension model is pointwise definable, meaning that every element init is definable without parameters. This is accomplished by adding to T theinfinitary assertions ∀ z W ϕ ∀ x ( ϕ ( x ) ⇐⇒ x = z ), taking the disjunct overall first-order formulas ϕ . These assertions ensure that every z is defined bya first-order formula, and the point is that the σ a arising in the proof canbe taken not only from L M , but also from amongst the definable elementsof L M , since these constitute an elementary substructure of L M .Remarkably, the theorem is true even for nonstandard models M , butthe proof above requires modification, since the infinitary deductions of M may not be well-founded deductions, and this prevents the use of soundnessto achieve the final contradiction. (One can internalize the contradiction tosoundness, if M should happen to have an uncountable L β | = ZFC , oreven merely arbitrarily large such β below ( ω L ) M .) To achieve the generalcase, however, Barwise uses his compactness theorem [1] and the theory ofadmissible covers to replace the ill-founded model M with a closely relatedadmissible set in which one may find the desired well-founded deductionsand ultimately carry out an essentially similar argument. I refer the readerto the accounts in [2] and in [3] .It turns out, however, that one does not need this extra technologyin the case of an ω -nonstandard model M of ZF , and so let me ex-plain this case. Suppose that M = h M, ∈ M i is an ω -nonstandard modelof ZF . Let T again be the theory ZFC + σ a for a ∈ M , where again σ a = ∀ z ( z ∈ ˇ a ⇐⇒ W b ∈ M a z = ˇ b ). Suppose there is no model of T + V = L .Consider the nonstandard theory ZFC M , which includes many nonstandardformulas. By the reflection theorem, every finite collection of ZFC axiomsis true in arbitrarily large L M β , and so by overspill there must be a non- uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L16
J. D. Hamkins standard finite theory
ZFC ∗ in M that includes every standard ZFC axiomand which M believes to hold in some L M β for some uncountable ordinal β in M . Let T ∗ be the theory ZFC ∗ plus all the σ a for a ∈ M . This theoryis Σ definable in M , and I claim that M must have a proof of a contra-diction from T ∗ + V = L in the infinitary logic L M∞ ,ω . If not, then the sameHenkin construction as above still works, working with nonstandard formu-las inside M , and the corresponding Henkin model satisfies all the actual(well-founded) assertions in T ∗ + V = L , which includes all of T + V = L , con-tradicting our preliminary assumption. So M has a proof of a contradictionfrom T ∗ + V = L . Since the assertion that there is such a proof is Σ , weagain find a proof in L M and even in L Mω . But we may now appeal to thefact that M thinks L Mβ is a model of ZFC ∗ plus σ a for every a ∈ L Mω , whichcontradicts the soundness principle of the infinitary deduction system in-side M . The point is that even though the deduction is nonstandard, thisdoesn’t matter since we are applying soundness not externally but inside M . The contradiction shows that T + V = L must have a model after all,and so M has an end-extension satisfying ZFC + V = L , as desired. Further-more, we may also ensure that every element of M becomes countable inthe end-extension as before.Let me conclude this section by mentioning another sense in which everycountable model of set theory is compatible in principle with V = L . Theorem 3.7: (Hamkins [6] ) Every countable model of set theory h M, ∈ M i , including every transitive model, is isomorphic to a submodelof its own constructible universe h L M , ∈ M i . In other words, there is anembedding j : M → L M , which is elementary for quantifier-free assertions. L M j M x ∈ y ←→ j ( x ) ∈ j ( y )Another way to say this is that every countable model of set theory isa submodel of a model isomorphic to L M . If we lived inside M , then byadding new sets and elements, our universe could be transformed into acopy of the constructible universe L M . uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L
4. An upwardly extensible concept of set
I would like now to explain how the mathematical facts identified in theprevious section weaken support for the V = L via maximize position,particularly for those set theorists inclined toward a pluralist or multiverseconception of the subject.To my way of thinking, theorem 3.2 already provides serious resistanceto the V = L via maximize argument, even without the multiverse ideas Ishall subsequently discuss. The point is simply that much of the force andcontent of large cardinal set theory, presumed lost under V = L , is never-theless still provided when the large cardinal theory is undertaken merelywith countable transitive models, and theorem 3.2 shows that this can bedone while retaining V = L . We often regard a large cardinal argument orconstruction as important—such as Baumgartner’s forcing of PFA over amodel with a supercompact cardinal—because it helps us to understand agreater range for set-theoretic possibility. The fact that there is indeed anenormous range of set-theoretic possibility is the central discovery of thelast half-century of set theory, and one wants a philosophical account ofthe phenomenon. The large cardinal arguments enlarge us by revealing theset-theoretic situations to which we might aspire. Because of the Baum-gartner argument, for example, we may freely assert
ZFC + PFA with thesame gusto and confidence that we had for
ZFC plus a supercompact car-dinal, and furthermore we gain detailed knowledge about how to transforma universe of the latter theory to one of the former and how these worldsare related. d Modifications of that construction are what led us to worldswhere MM holds and MM + and so on. From this perspective, a large partof the value of large cardinal argument is supplied already by our abilityto carry it out over a transitive model of ZFC , rather than over the fulluniverse V .The observation that we gain genuine set-theoretic insights when work-ing merely over countable transitive models is reinforced by the fact thatthe move to countable transitive models is or at least was, for many settheorists, a traditional part of the official procedure by which the forcingtechnique was formalized. (Perhaps a more common contemporary view isthat this is an unnecessary pedagogical simplification, for one can formal-ize forcing over V internally as a ZFC construction.) Another supporting d The converse question, however, whether we may transform models of
PFA to modelsof
ZFC + ∃ supercompact cardinal, remains open. Many set theorists have conjecturedthat these theories are equiconsistent.uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L18
J. D. Hamkins example is provided by the inner model hypothesis of [4], a maximality-type principle whose very formalization seems to require one to think ofthe universe as a toy model, for the axiom is stated about V as it exists asa countable transitive model in a larger universe. In short, much of whatwe hope to achieve with our strong set theories is already achieved merelyby having transitive models of those theories, and theorem 3.2 shows thatthe existence of any and all such kind of transitive models is fully andequally consistent with our retaining V = L . Because of this, the V = L via maximize argument begins to lose its force.Nearly every set theorist entertaining some strong set-theoretic hypoth-esis ψ is generally also willing to entertain the hypothesis that ZFC + ψ holds in a transitive model. To be sure, the move from a hypothesis ψ tothe assertion ‘there is a transitive model of ZFC + ψ ’ is strictly increasingin consistency strength, a definite step up, but a small step. Just as philo-sophical logicians have often discussed the general principle that if you arewilling to assert a theory T , then you are also or should also be willingto assert that ‘ T is consistent,’ in set theory we have the similar principle,that if you are willing to assert T , then you are or should be willing toassert that ‘there is a transitive model of T ’. What is more, such a prin-ciple amounts essentially to the mathematical content of the philosophicalreflection arguments, such as in [18], that are often used to justify largecardinal axioms. As a result, one has a kind of translation that maps anystrong set-theoretic hypothesis ψ to an assertion ‘there is a transitive modelof ZFC + ψ ’, which has the same explanatory force in terms of describingthe range of set-theoretic possibility, but which because of the theorems ofsection 3 remains compatible with V = L .This perspective appears to rebut Steel’s claims, mentioned in the open-ing section of this article, that “there is no translation” from the large car-dinal realm to the V = L context and that “adding V = L ...prevents usfrom asking as many questions.” Namely, the believer in V = L seems fullyable to converse meaningfully with any large cardinal set theorist, simplyby imagining that the large cardinal set theorist is currently living inside acountable transitive model. By applying the translation ψ ‘there is a transitive model of ZFC + ψ ’ , the V = L believer steps up in strength above the large cardinal set theorist,while retaining V = L and while remaining fully able to analyze and carryout the large cardinal set theorist’s arguments and constructions inside thattransitive model. Furthermore, if the large cardinal set theorist believes in uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L her axiom because of the philosophical reflection principle arguments, thenshe agrees that set-theoretic truth is ultimately captured inside transitivesets, and so ultimately she agrees with the step up that the V = L believermade, to put the large cardinal theory inside a transitive set. This simplyreinforces the accuracy with which the V = L believer has captured thesituation.Although the translation I am discussing is not a ‘fair interpretation’in the technical sense of [16], as discussed in section 2, nevertheless it doesseem to me to be a fair interpretation in a sense that matters, because itallows the V = L believer to understand and appreciate the large cardinalset theorist’s arguments and constructions.Let me now go a bit further. My claim is that on the multiverse view as Idescribe it in [9] (see also [5 , , V is revealed in part by the toy simulacrum of it that we find amongst thecountable models of set theory. For all we know, our current set-theoreticuniverse V is merely a countable transitive set inside another much largeruniverse V + , which looks upon V as a mere toy. And so when we can provethat a certain behavior is pervasive in the toy multiverse of any modelof set theory, then we should expect to find this behavior also in the toymultiverse of V + , which includes a meaningfully large part of the actualmultiverse of V . In this way, we come to learn about the full multiverseof V by undertaking a general study of the toy model multiverses. Justas every countable model has actual forcing extensions, we expect our fulluniverse to have actual forcing extensions; just as every countable modelcan be end-extended to a model of V = L , we expect the full universe V can be end-extended to a universe in which V = L holds; and so on. Howfortunate it is that the study of the connections between the countablemodels of set theory is a purely mathematical activity that can be carriedout within our theory. This mathematical knowledge, such as the resultsmentioned in section 3 or the results of [5], which show that the multiverseaxioms of [9] are true amongst the countable computably-saturated modelsof set theory, in turn supports philosophical conclusions about the natureof the full set-theoretic multiverse.The principle that pervasive features of the toy multiverses are evidencefor the truth of those features in the full multiverse is a reflection principlesimilar in kind to those that are often used to provide philosophical jus-tification for large cardinals. Just as those reflection principles regard thefull universe V as fundamentally inaccessible, yet reflected in various muchsmaller pieces of the universe, the principle here regards the full multiverse uly 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L20
J. D. Hamkins as fundamentally inaccessible, yet appearing in part locally as a toy mul-tiverse within a given universe. So our knowledge of what happens in thetoy multiverses becomes evidence of what the full multiverse may be like.Ultimately, the multiverse vision entails an upwardly extensible con-cept of set, where any current set-theoretic universe may be extended toa much larger, taller universe. The current universe becomes a countablemodel inside a larger universe, which has still larger extensions, some withlarge cardinals, some without, some with the continuum hypothesis, somewithout, some with V = L and some without, in a series of further exten-sions continuing longer than we can imagine. Models that seem to have 0 ♯ are extended to larger models where that version of 0 ♯ no longer works as0 ♯ , in light of the new ordinals. Any given set-theoretic situation is seenas fundamentally compatible with V = L , if one is willing to make themove to a better, taller universe. Every set, every universe of sets, becomesboth countable and constructible, if we wait long enough. Thus, the con-structible universe L becomes a rewarder of the patient , revealing hiddenconstructibility structure for any given mathematical object or universe, ifone should only extend the ordinals far enough beyond one’s current set-theoretic universe. This perspective turns the V = L via maximize argu-ment on its head, for by maximizing the ordinals, we seem able to recover V = L as often as we like, extending our current universe to larger andtaller universes in diverse ways, attaining V = L and destroying it in anon-again, off-again pattern, upward densely in the set-theoretic multiverse,as the ordinals build eternally upward, eventually exceeding any particularconception of them. References
1. Jon Barwise. Infinitary logic and admissible sets.
J. Symbolic Logic ,34(2):226–252, 1969.2. Jon Barwise. Infinitary methods in the model theory of set theory. In
LogicColloquium ’69 (Proc. Summer School and Colloq., Manchester, 1969) , pages53–66. North-Holland, Amsterdam, 1971.3. Jon Barwise.
Admissible sets and structures . Springer-Verlag, Berlin, 1975.An approach to definability theory, Perspectives in Mathematical Logic.4. Sy-David Friedman. Internal consistency and the inner model hypothesis.
Bull. Symbolic Logic , 12(4):591–600, 2006.5. Victoria Gitman and Joel David Hamkins. A natural model of the multiverseaxioms.
Notre Dame J. Form. Log. , 51(4):475–484, 2010.6. Joel David Hamkins. Every countable model of set theory embeds into itsown constructible universe. pages 1–26. under review. uly 27, 2018
Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in
Multiverse˙on˙V=L
Multiverse perspective on V = L
7. Joel David Hamkins. Is the dream solution of the continuum hypothesis at-tainable? pages 1–10. under review.8. Joel David Hamkins. The set-theoretic multiverse : A natural context for settheory.
Annals of the Japan Association for Philosophy of Science , 19:37–55,2011.9. Joel David Hamkins. The set-theoretical multiverse.
Review of SymbolicLogic , 5:416–449, 2012.10. Joel David Hamkins, David Linetsky, and Jonas Reitz. Pointwise definablemodels of set theory. to appear in Journal of Symbolic Logic .11. Daniel Isaacson. The reality of mathematics and the case of set theory. InZsolt Novak and Andras Simonyi, editors,
Truth, Reference, and Realism .Central European University Press, 2008.12. Benedikt L¨owe. A first glance at non-restrictiveness.
Philosophia Mathemat-ica , 9(3):347–354, 2001.13. Benedikt L¨owe. A second glance at non-restrictiveness.
Philosophia Mathe-matica , 11(3):323–331, 2003.14. Penelope Maddy. Believing the axioms, I.
The Journal of Symbolic Logic ,53(2):481–511, 1988.15. Penelope Maddy. Believing the axioms, II.
The Journal of Symbolic Logic ,53(3):736–764, 1988.16. Penelope Maddy. V = L and MAXIMIZE. In Logic Colloquium ’95 (Haifa) ,volume 11 of
Lecture Notes Logic , pages 134–152. Springer, Berlin, 1998.17. Donald A. Martin. Multiple universes of sets and indeterminate truth values.
Topoi , 20(1):5–16, 2001.18. W. N. Reinhardt. Remarks on reflection principles, large cardinals, andelementary embeddings.
Proceedings of Symposia in Pure Mathematics ,13(II):189–205, 1974.19. Saharon Shelah. Logical dreams.
Bulletin of the American Mathematical So-ciety , 40:203–228, 2003.20. John R. Steel. Generic absoluteness and the continuum problem. Slidesfor a talk at the Laguna workshop on philosophy and the contin-uum problem (P. Maddy and D. Malament organizers) March, 2004.http://math.berkeley.edu/ ∼∼