Mathematical Pull

Colin J. Rittberg

Draft, not to be circulated

forthcoming

In this paper I show that mathematicians can successfully engage inmetaphysical debates by mathematical means. I present the contem-porary work of Hugh Woodin and Peter Koellner. Woodin has pro-posed intrinsically appealing axiom-candidates which could, whenadded to our current set theoretic axiom system, resolve the issuethat some fundamental questions of set theory are formally unsolv-able. The proposed method to choose between these axioms is torely on future results in formal set theory. Koellner connects thisto a contemporary metaphysical debate on the ontological nature ofsets. I argue, mathematics is connected to the philosophical debatein such a way that by doing more mathematics an argument in thephilosophical debate can be obtained. This story reveals an activeconnectedness between mathematics and philosophy.

1 Introduction

There is a connection between mathematics and metaphysics. Some mathemati-cians have told us that their metaphysical views influence their mathematicalthought. Here is the well-known quote from Godel on this matter:

[M]y objectivistic conception of mathematics and metamathematicsin general, and of transfinite reasoning in particular, was fundamen-tal also to my other work in logic (A logical journey: From Godel toPhilosophy, p. 241).

Godel tells us that metaphysics can influence mathematics. Conversely, math-ematics can also influence metaphysics. For example, in her Defending theAxioms Maddy takes the practice of set theory seriously– this is her secondphilosophical approach– and draws metaphysical conclusions from her researchabout this practice; [Ma2011]. Notice that in both Godel’s and Maddy’s casethe influencee is not working on the discipline that does the influencing. Theseare hence cases of passive influence between mathematics and metaphysics. Inthis paper, I argue that mathematics can actively influence metaphysics. I callthis kind of influence mathematical pull.

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In the above paragraph, a certain understanding of the term ‘mathemat-ics’ is at play, which takes the practice of mathematics seriously. Heuristicprinciples play a role in the practice of mathematics,1 and this is why heuris-tic principles are considered as part of mathematics in the above. To this, aLakatosian formalist would disagree. Lakatosian formalists ‘identif[y] mathe-matics with its formal axiomatic abstraction’ ([La1976], p. 1) and thereforehold that heuristic principles are not part of mathematics. The study of math-ematical practice however has given us ample evidence that heuristics are infact used in mathematics as practised. The Lakatosian formalist’s story aboutmathematics therefore does not account for mathematics as practised, which iswhy I will not consider Lakatosian formalism any further in this paper.

In this paper, I show that mathematicians are actively influencing meta-physics by mathematical means; mathematicians establish mathematical pull.I present a story in which mathematicians connect mathematics to a meta-physical debate in such a way that future mathematical results may decide themetaphysical debate. The story is a story about contemporary set theory. Con-temporaneity allows for a story which highlights how mathematics is set up topull the metaphysical debate by future results in mathematics, and this showsthat mathematicians are actively looking for mathematical pull. Set theory is aparticularly fitting discipline to take as an example because of the belief sharedby many philosophers that in order to understand the ontological nature ofmathematics, we need to understand the ontological nature of sets. What thestory I am about to tell shows is that the mathematicians are also trying tounderstand the ontological nature of sets, and they have reasonable success inthis. The philosophers should therefore pay close attention to the story and thedirection in which the future mathematical results will pull the metaphysicaldebate.

2 The Story

In this section I tell the story about Koellner’s active search for mathematicalpull. I start, in 2.1, by delineating the metaphysical problem. In 2.2 I thenpresent the mathematics which Koellner connects to the metaphysical problemto establish mathematical pull. In 3 I discuss what we can learn from this story.

2.1 The Problem

There are questions about sets which are formally unsolvable from our currentlyaccepted ZFC axioms. A well-known example of such an unsolvable questionis Cantor’s question about the size of the continuum: is there an uncountable

1Here are four examples of philosophers who have argued for the above point. Pene-lope Maddy analyses reasoning in set theory and discusses two heuristic principles in detail;[Ma1997]. Dirk Schlimm tells us about the creative potential of axioms in [Sch2009]. KennethManders discusses diagram-based reasoning in [Man2007]. Stav Kaufman has a detailed storyabout surprise-into-story-into-condition style arguments; see this volume. A surprise-into-story-into-condition style argument will appear later on in this paper.

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subset of R which has cardinality less than the cardinality of R? Cantor hy-pothesised that there is no such uncountable set. This hypothesis has becomeknown as the Continuum Hypothesis, CH. In 1963 Paul Cohen’s results, inconjunction with earlier results due to Kurt Godel, established that the CH isneither provable nor disprovable from the ZFC axioms; it is independent.

Cantor’s Continuum problem is not the only problem which is undecidablefrom ZFC. Woodin sees set theory haunted by a ‘spectre of undecidability’([Wo2010a], p. 17). This is a problem for him, which he has called the ZFCdilemma:

The ZFC Dilemma: Many of the fundamental questions of SetTheory are formally unsolvable from ZFC axioms. ([Wo2009a], p.1)

Notice that Woodin does not intend ‘dilemma’ here as a choice between two ormore undesirable alternatives but rather in its informal meaning as ‘a difficultproblem or situation’, as becomes clear from his formulation of the dilemma.2

Why is this a difficult situation? Because by the time of writing the abovecited paper Woodin had become a non-pluralist,3 i.e. for Woodin there is anobjective universe of sets which set theorists discover. From this point of view,the challenge of this ‘dilemma’ is that the independence results show a povertyof our currently available formal methods of discovery.

Conversely, the pluralist holds that there is no single universe of sets butmany distinct universes of sets. Different universes have different set-theoreticstatements that hold in them; e.g. there are set-theoretic universes in whichCH holds and those in which CH fails. Committing to a multitude of uni-verses hence means that there is not one correct concept of set-hood, but manydifferent, possibly contradicting concepts. A noteworthy proponent of contem-porary pluralism is the set theorist Joel David Hamkins; [Ha2011], [Ha2012].For Hamkins, the independence results are no dilemma for set theory, in fact,he uses them to argue in favour of his version of pluralism.

The pluralism/non-pluralism debate originates in mathematical practice, buthas become a metaphysical debate about how many correct concepts of set-hoodthere are; [Ko2014], [Wo2009a], [Ha2011], [Mag2012]. This is the debate towhich Koellner connects mathematics in such a way that it can pull the debate.The idea is that the non-pluralists could strengthen their case if they couldresolve the ZFC dilemma and somehow banish the ‘spectre of undecidability’.And Woodin has presented a way how this could be done.

2Notice that if ZFC were inconsistent, then every foral set-theoretical statement could beproven from ZFC and hence there would be no dilemma. Therefore Woodin seems to assumethe consistency of ZFC here. I will do the same throughout this paper.

3Woodin calls this view the ‘set theorist’s view’; [Wo2009a], [Wo2009b]. I prefer the term‘non-pluralism’ because is better captures the idea behind the view. This term stems fromKoellner, who defines it in terms of belief in the existence of theoretical solutions to theformally unsolvable questions of set theory; [Ko2014]. He has an argument for ontologicalcommitment attached to this view which I sidestepped through my definition. The resultingphilosophical differences are minor and play no role in the argument of this paper.

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In what follows I present Woodin’s work towards a resolution to the ZFCdilemma. His work has led to a presentation of a family of axioms which haveintrinsic appeal and which would also settle the issue of undecidability of the(contemporary) fundamental questions of set theory. Because some undecidableproblems are differently answered by different axioms of the discussed kind,most notably the CH, the non-pluralists will have to settle for just one ofthese axioms. Koellner proposes to analyse the implications of these axioms inmore detail. These future results can pull the pluralism/non-pluralism debate;Koellner establishes mathematical pull. The idea is that if these future resultslead to a certain type of convergence under a given axiom, then this strengthensthe case for the axiom to be a correct description of the universe of sets andhence pulls us towards non-pluralism. However, it could happen that there willbe no convergence of the desired type under any of the axioms. This would pullus towards pluralism.

The concept of a large cardinal plays a crucial role in Woodin’s proposal. Iexplain the concept in 2.2.1. The next step, in 2.2.2 and 2.2.3, is to reconstructWoodin’s argument which leads to the family of axioms which could resolvethe ZFC dilemma. The ‘spectre of undecidability’ resurfaces at this point be-cause of a choice-problem between the different axioms, and I present, in 2.2.4,the proposal to decide on the basis of a certain type of convergence. Koellnerconnects these convergence considerations with the pluralism/non-pluralism de-bate, thereby successfully establishing mathematical pull. Importantly, we donot yet know how the convergence results will turn out, and therefore the storyshows that Koellner is actively searching to decide the metaphysical debateby mathematical means; he is actively searching for mathematical pull. Thesematter are further discussed in 3.

2.2 The mathematics

This subsection starts, in 2.2.1, with a short review of some of the features oflarge cardinal axioms and Woodin’s argument for the truth of these axioms.Work on the large cardinal axioms is intimately connected to the inner modelprogramme, which I will explain in 2.2.2. Woodin’s argument relies on a newresult in the inner model programme, which I will explain in 2.2.3. At the endof this passage the ‘spectre of undecidability’ resurfaces. In 2.2.4 I present theconvergence argument, which could resolve the undecidability issue. As a secondstep in this passage, I explain Koellner’s proposal to connect the convergenceargument to the pluralism/non-pluralism debate.

I do not discuss the technicalities of large cardinal theory in what follows;see [Je2006] and [Ka2009] for details. The latter also elaborates on the historyof large cardinals. The original papers by Woodin and Koellner which relateto the topic of this section are [Wo2009a], [Wo2010a], [Wo2009b], [Ko2014] and[Ko2013a]. For an historical development of Woodin’s thought see [Ri2014].

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2.2.1 Large Cardinal Axioms

In his canonical book on large cardinals, The Higher Infinite, Kanamori writesabout large cardinal axioms:

These hypotheses posit cardinals that prescribe their own transcen-dence over smaller cardinals and provide a superstructure for theanalysis of strong propositions. As such they are the rightful heirsto the two main legacies of Georg Cantor, founder of set theory: theextension of number into the infinite and the investigation of defin-able sets of reals. The investigation of large cardinal hypotheses isindeed a mainstream of modern set theory. ([Ka2009], p. XI)

A large cardinal axiom then is a statement which affirms the existence of a set,the large cardinal. These large cardinals are such that they ‘prescribe their owntranscendence over smaller cardinals’, which is why they are also called ‘strongaxioms of infinity’. For example, one can consider the Axiom of Infinity as alarge cardinal axiom because it affirms the existence of a set (namely ω) whichis transcendent over all those cardinals which can be proven to exist withoutthe Axiom of Infinity (in this case: over all finite cardinals). Similarly with theAxiom of Replacement. Without this axiom no sets of cardinality ℵω or biggercan be proven to exist. These two axioms however have already been acceptedinto our contemporary axiomatic system, whereas other large cardinal axiomsdo not hold this place of honour. The Axiom of Infinity and the Axiom ofReplacement are usually not seen as large cardinal axioms in a formal setting.For example, Kanamori, in his ‘chart of cardinals’ ([Ka2009], p. 472), lists 28different types of large cardinal axioms, and neither the Axiom of Infinity northe Axiom of Replacement is on this list.

One well-known large cardinal axiom which does appear on Kanamori’s listasserts the existence of an inaccessible cardinal, i.e. an uncountable regular4

cardinal κ such that for all λ < κ: 2λ < κ. The existence of an inaccessiblecardinal cannot be proven from ZFC. To see this consider the von Neumanhierarchy

V0 = ∅Vα+1 = P(Vα)

Vα =⋃β<α

Vβ ; α ∈ LIM

It follows from these definitions that if κ is an inaccessible cardinal, then Vκis a model of ZFC. This means that ZFC+‘there is an inaccessible cardinal’proves the consistency of ZFC, which is unprovable from ZFC alone. HenceZFC+‘there is an inaccessible cardinal’ is stronger than ZFC in this sense.

Kanamori lists 28 different types of large cardinal axioms in his chart anda development of all of them is well beyond the scope of this paper. Instead, I

4‘Regular’ is a technical term, meaning that for every λ < κ there is no co-final functionfrom λ to κ, i.e. there is no f : λ→ κ such that ran(f) is unbounded in κ.

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briefly suggest why large cardinal axioms are mathematically interesting. I willnot give any formal definitions of the axioms mentioned as there is little philo-sophical advantage in doing so; see [Ka2009] or [Je2006] for these definitions.Once I have suggested why large cardinals are mathematically interesting I ex-plain a fact about them, namely that they give rise to degrees of unsolvability.This fact is used by Woodin to argue for the truth of the large cardinal axioms,and I discuss his argument at the end of this subsection. Furthermore, degreesof unsolvability rely on the large cardinal hierarchy, explained below, which isimportant for the inner model programme presented in 2.2.2.

Let me begin by suggesting why large cardinals are mathematically inter-esting. Large cardinal axioms can partly resolve the ZFC dilemma. The aboveexample of an inaccessible cardinal is such a case. Adding the axiom ‘thereis an inaccessible cardinal’ to ZFC results in a new theory which can proveCon(ZFC), i.e. the statement expressing the consistency of ZFC. A problemthat is formally unsolvable in ZFC (namely ‘does Con(ZFC) hold?’) becomessolvable in a large cardinal extension. There are many more examples of prob-lems which are formally unsolvable from ZFC but can be solved in a largecardinal extension. For example Lebesgue’s measure problem5 cannot be solvedfrom the axioms of ZFC alone, but adding the axiom ‘there is a measurablecardinal κ’ gives us a large cardinal extension in which the problem is solved(for cardinals up to and including κ). Similarly, the Banach-Tarski paradox canbe tamed6 by moving to a large cardinal extension. In this sense, large cardinalaxioms are powerful: they resolve the ZFC dilemma in many cases.

There are limitations to the power of large cardinals; not all formally unsolv-able problems can be solved by large cardinal axioms. The well-known exampleis the Continuum Hypothesis. Levy and Solovay have shown in their [LeSo1967]that no addition of a large cardinal axiom to ZFC can prove or disprove theCH. This shows that even if we assume large cardinal axioms, there remainformally unsolvable problems in set theory. In this sense the large cardinal ax-ioms do not fully banish the ‘spectre of undecidability’ from contemporary settheory and hence do not fully resolve the ZFC dilemma.

I now explain how large cardinal axioms give rise to degrees of unsolvability,a fact I which will use throughout the remainder of this section. It starts withan ordering of the large cardinals by consistency strength. This consistencystrength order is defined as follows: a large cardinal axiom φ is stronger in the

5For the real line the measure problem is the question whether there is a measure on thereals, i.e. whether there is a function m from all bounded sets of reals to the non-negativereals such that m is translation invariant, countably additive and not identical to zero. Noticethat in ZFC the Lebesgue measure does not solve the problem due to the Vitali set. Itwas eventually solved by Ulam; see [Ka2009], pp. 22-27 for a historical as well as technicaldiscussion.

6Recall that Banach and Tarski showed that a three-dimensional sphere can be partitionedin such a way that, through rotations and translations (i.e. without changing the size ofthe pieces), two spheres can be obtained which are identical with the first. This paradoxicalpartition is a direct implication of the Axiom of Choice. By assuming certain large cardinals(namely: infinitely many Woodin cardinals) one can ensure that the Axiom of Choice is usedin a strong sense for these partitions (namely: the partition must be complicated, no projectivesets suffice). See [Wo2001a] for a further discussion.

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consistency strength order than a large cardinal axiom ψ if it is provable in ZFCthat Con(ZFC + φ) implies Con(ZFC + ψ). Interestingly, this forms a linearorder of the large cardinal axioms, i.e. for any two large cardinal axioms oneis stronger than the other. Hence, the large cardinals form a linearly orderedhierarchy.

Recall that some problems which are formally unsolvable from ZFC alonecan be solved in certain large cardinal extensions. By discovering which largecardinal axiom is necessary and sufficient to solve the problem, the problem iscalibrated in the large cardinal hierarchy. The degree of unsolvability of theproblem is hence the large cardinal axiom which is needed to solve the problem.That this really is a ‘degree’ and hence comparable to the degrees of unsovabilityof other problems follows from the fact that the large cardinals form a hierarchy:by calculating the degree of unsolvability of a problem we effectively positionit in the large cardinal hierarchy. But large cardinals are comparable via theconsistency strength order. Hence degrees of unsolvability are comparable. Thismeans that large cardinals allow us to compare seemingly unconnected problemsby calculating their degrees of unsolvability.

We have seen that there are problems of contemporary set theory for whicha degree of unsolvability cannot be calculated; the CH was the prominent ex-ample. Hence, large cardinal axioms help resolving Woodin’s ZFC dilemma,but they do not resolve it completely.

Woodin claims that large cardinal axioms are ‘true axioms about the universeof sets’ ([Wo2009a], p. 5). The rest of this subsection is devoted to his argumentfor this position.

It all starts with Woodin’s prediction about the consistency of a certain the-ory, call it theory T . The precise nature of T is technical and not illuminatingfor the point I wish to make; see [Wo2009a], pp. 2-5, for details.7 The con-sistency of T cannot be proven from ZFC (assuming that ZFC is consistent).Woodin reminds us that if T were inconsistent, then the inconsistency could beproven in finitely many steps. However, there is currently no indication thatthis is the case. With this in mind, consider the following prediction:

There will be no discovery ever of an inconsistency in [T ]. ([Wo2009a],p. 6, emphasis in original)

Woodin calls this a ‘specific and unambiguous prediction about the physicaluniverse’ (ibid.) which could be refuted by ‘finite evidence’ ([Wo2009a], p. 5).Therefore, so Woodin holds, ‘[o]ne can arguably claim that if this [...] predictionis true, then it is a physical law’ ([Wo2009a], p. 6). Woodin holds that if thisprediction is true (and currently we have no reason to believe that it is not),then set theorists should be able to account for it. For Woodin the way to doso is via large cardinals. It is a formally proven fact about theory T that it isconsistent if and only if the large cardinal extension ZFC + ‘there are infinitelymany Woodin cardinals’ is. Using this information, Woodin writes about theabove prediction:

7For connoisseurs: the theory T is ZFC + SBH, whereby SBH denotes the stationarybasis hypothesis.

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It is through the calibration by a large cardinal axiom in conjunc-tion with our understanding of the hierarchy of such axioms astrue axioms about the universe of sets that this prediction isjustified. ([Wo2009a], p. 5, emphasis in original)

Woodin says ‘[a]s a consequence of my belief in this claim, I make [the above]prediction’ (ibid.). Hence, Woodin accounts for the prediction by a) the calibra-tion of the problem of consistency of T by large cardinal axioms and b) the truthof the large cardinal axioms. Part a) is a formal result, what is at stake here ispart b). If b) is needed to make the prediction, then we have reason to believe inb). Assume that b) does not hold. Then all we have is an equiconsistency resultof the theory T with ZFC + ‘there are infinitely many Woodin cardinals’. But,Woodin claims, ‘[j]ust knowing the [...] two theories are equiconsistent does notjustify [the] prediction at all’ (ibid.). Hence we need b) to make the prediction,i.e. we have reason to believe that large cardinal axioms are true.

My focus in this paper is Koellner’s mathematical approach to thepluralism/non-pluralism debate. Therefore, I will not critically assess Woodin’sargument for the truth of the large cardinal axioms, with one exception. Itseems possible for the pluralist to hold that a set-theoretic universe exists (ina strong sense) in which there are infinitely many Woodin cardinals. This uni-verse could then serve to justify Woodin’s prediction, without committing tothe position that the relevant large cardinal axiom is true (it is only true in arelative sense, namely relative to the model). Hence Woodin seems to assumenon-pluralism in his argument for the truth of large cardinal axioms.8 I willrefer to this feature of Woodin’s argument in the final section of this paper.

2.2.2 The Inner Model Programme

Woodin’s development of the axioms that could banish the ‘spectre of undecid-ability’ and which are connected, by Koellner’s argument, to the pluralism/non-pluralism debate, originates in the inner model programme. In this subsectionI present those features of the inner model programme which are important fora philosophical understanding of Woodin’s development of his axioms.

The aim of the inner model programme is to construct mathematically ac-cessible structures which can accommodate large cardinals. This programmetook off from considerations about Godel’s L. Recall that Godel had assumedthat ZF (i.e. ZFC without the Axiom of Choice) is consistent and constructed

8It should be noted that Woodin has a separate argument against pluralism; [Wo2009b],pp. 16-20. However, the pluralism Woodin criticises is not the form of pluralism which is,for example, held by Hamkins. As such it is of less interest and will not be discussed in thispaper.

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from there his constructible universe L as follows:

L0 = ∅Lα+1 = Def(Lα)

Lα =⋃β<α

Lβ ; α ∈ LIM

and L is the union over all Lα andDef(X) denotes all the definable subsets ofX.From this definition it follows that all axioms of ZFC hold in L. Furthermore,L is mathematically accessible in the sense that set theorists can successfullystudy this structure. Today the structure theory of L, i.e. those statementsthat can be proven to hold in L, is very well understood.

With respect to large cardinals, there is a problem. In 1961 Scott showed thatif there are measurable cardinals, then L is not the whole universe of sets. Sincemeasurable cardinals are relatively low in the hierarchy of large cardinals, Scott’stheorem shows that L does not accommodate any large cardinals which areeven only moderately high up in the hierarchy. Therefore, to have an accessiblestructure which can accommodate large cardinals on the higher levels of thelarge cardinal hierarchy, new models need to be constructed. The search forthese new models forms the inner model programme.

The methodology of the inner model programme is based on Godel’s originalmethodology. I have mentioned that Godel assumed ZF to be consistent andthen constructed a model for this theory. In the inner model programme oneassumes ZFC + LCA to be consistent (for some targeted large cardinal axiomLCA) and tries to construct a model for this theory.

Here is an example of an inner model construction.9 To construct an in-ner model for the axiom ‘there is a measurable cardinal’ one assumes thatZFC+‘there is a measurable cardinal’ is consistent. The existence of the mea-surable cardinal can then be coded into an extender E, a certain type of function.One can now construct L but instead of allowing at the step Lα+1 only param-eters from Lα in the defining formulae one now allows parameters from E aswell. This leads to the following construction

L0[E] = ∅

Lα+1 = Def(Z), where

Z = Lα[E] ∪ {E ∩ Lα[E]}

and Def(Z) refers to the definable powerset of Z.

Lα =⋃{Lβ | β < α} for α limit ordinal.

Now denote by L[E] the class of all sets a such that a ∈ Lα[E] for some α.From these definitions one can prove that L[E] is a model of ZFC+‘there is

9For a more detailed account, see e.g. [Wo2009a]. See [Ka2009] for further technical issues.

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a measurable cardinal’. Hence L[E] can accommodate a measurable cardinal.Its definition also makes it accessible to mathematical study. Hence, L[E] is aswanted and we call it an inner model for the theory ZFC+‘there is a measurablecardinal’.

Recall that L was limiting in the sense that it does not accommodate anylarge cardinal at the level of a measurable or above. The above L[E] is limiting ina similar fashion. One can show that it cannot accommodate any large cardinalaxiom stronger than ‘there is a measurable cardinal’. It is hence up to the innermodel programme again to construct models for these stronger large cardinals.However, all these inner models come with such limiting results: only largecardinals up to the level of the targeted large cardinal can be accommodated,but no stronger ones. Thus far no ultimate inner model has been constructedwhich can accommodate all large cardinals (but Woodin argues that it mightbe possible, see next subsection). Currently the strongest large cardinal axiomfor which an inner model has been constructed states the existence of a Woodincardinal which is the limit of Woodin cardinals; [Wo2009a], p. 15. These largecardinals are slightly above the middle of the large cardinal hierarchy.

If one could push the inner model programmes beyond its current limits andconstruct an inner model for a supercompact cardinal, a large cardinal strongerthan any large cardinal for which an inner model has thus far been constructed,then something remarkable happens. This is the topic of the next subsection.

2.2.3 Woodin’s New result

In this subsection I present a new and remarkable result Woodin was able toprove. This result leads to the family of axioms which, by Koellner’s argument,are connected to the pluralism/non-pluralism debate.

The gist of Woodin’s new result is that it might be possible to rid the innermodel programme of the limiting results. Recall here that the construction ofan inner model was such that only the targeted large cardinal axiom but nostronger one can be accommodated in the constructed inner model. Woodin’snew result is that this changes at the level of a supercompact cardinal.

Woodin considers what would happen if an inner model which accommo-dates a supercompact cardinal could be found. Assuming that there is an innermodel which can accommodate a supercompact cardinal, Woodin is able to showthat, ‘and this is the surprise’ ([Wo2009a], p. 20), unlike other models build inthe inner model programme this model would accommodate all large cardinalaxioms (consistent with ZFC)10. This point should be stressed. Usually, if wesucceed in building an inner model for some large cardinal axiom LCA, then nolarge cardinal axiom stronger than LCA holds in this model. But, and this isthe surprising part, if we succeed in building a model for a supercompact car-dinal, then this model, unlike the others, can accommodate all large cardinals,even those that are stronger than supercompact cardinals. For Woodin this isa ‘paradigm shift in the whole conception of inner models’ ([Wo2009a], p. 21).

10For now on I will use ‘all large cardinals’ to mean ‘all large cardinal axioms consistentwith ZFC’.

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An inner model which can accommodate a supercompact cardinal would bean ultimate step in the inner model programme of generalising L to accountfor more and more large cardinal axioms, precisely because it could accountfor all large cardinals. This is why Woodin has called such an ultimate innermodel Ultimate-L, written as Ult-L. Thus far, Woodin has been unable toconstruct an ultimate L; see concluding remarks of [Wo2009a] and [Wo2009b].Hence, we currently do not have a mathematically accessible structure whichcan accommodate all large cardinals.

At this point, Woodin uses what I have called the surprise-into-story-into-condition method. He is surprised that at a level of a supercompact cardinalinner models can accommodate all large cardinals. This surprise is now turnedinto a story about the universe of sets: the universe of sets is an ultimate-L. Thisstory then gives rise to a condition, a new form of axiom that we should accept.Before I say a bit more about these axioms, let me point out here that Kaufmanhas discussed the surprise-into-story-into-condition method; see her entry in thisvolume. The method hence seems to belong to ordinary mathematical practice.

There are various axiom candidates which could function as the axiom thatthe universe of sets is an ultimate-L, expressed as V = Ult-L; [Ko2013a].11

The various axiom candidates for V = Ult-L form the family of axioms whichKoellner uses to establish mathematical pull to the pluralism/non-pluralismdebate, which is why they will be at the center of attention for the rest of thispassage.

The V = Ult-L axioms are powerful. They would banish the ‘spectre ofundecidability’ because

There is no known candidate for a sentence which is independentfrom [a version of V = Ult-L] and which is not a consequence ofsome large cardinal axiom. ([Wo2009a], p. 27)

Hence, all contemporary fundamental questions are decidable in the large car-dinal extension ZFC + V = Ult-L+ LCA, where V = Ult-L stands for one ofthe versions of this axiom and LCA stands for a schema expressing that largecardinals exist. This result links backs to the discussion of the pluralism/non-pluralism debate. I had mentioned that the non-pluralists could strengthen theircase if they could find an axiom which would banish the ‘spectre of undecid-ability’. The versions of V = Ult-L (plus large cardinals) do precisely this job.Assuming ZFC + V = Ult-L+LCA there are no more contemporary formallyunsolvable questions, the ZFC dilemma is solved.

But yet again there is a problem: the different versions of V = Ult-L con-tradict each other. For example, there is a version which implies the CH andone which implies ¬CH. Hence, those non-pluralists who have subscribed tothe argument thus far are forced to choose between the different versions ofV = Ult-L, but how to do so? The ‘spectre of undecidability’ resurfaces herebecause it is, prima facie, unclear how our set theoretic methods of discovery

11For a presentation of one of the axioms for V = Ult-L and possibilities for its generalisa-tion, see [Wo2010a], p. 17.

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could deal with the issue. Koellner’s proposal to resolve the issue is presentedbelow.

2.2.4 Convergence

Above I introduced the V = Ult-L axioms, mentioned that they banish thecontemporary ‘spectre of undecidability’ and argued that the spectre resurfacesin the form of the different and mutually exclusive versions of V = Ult-L. In[Ko2013a], Koellner discusses a method that could lead to a decision between thedifferent V = Ult-L axioms. In this subsection I first explain this method andthen present how Koellner connects this method to the pluralism/non-pluralismdebate.

The proposed method to choose between the ultimate Ls is to analyse, foreach version of V = Ult-L, the structure theories of two set-theoretical struc-tures under the assumed axiom. One of these structures is well-known to theset theory community, the other has not received much attention yet. The ideais that if the structure theories of these two structures converge in similarityunder a version of V = Ult-L, then this counts as evidence for this version ofV = Ult-L. To elaborate on this I will therefore have to explain the two struc-tures that are to be analysed and give an idea of what is meant by the similarityof structure theories.

My elaboration starts with the structure which is well-known to the settheory community: L(R).12 This structure contains all the reals and all theirdefinable subsets. It ‘has figured prominently in the investigation of stronghypotheses’ ([Ka2009], p. 142) and set theorists have come to an intimateunderstanding of this structure.

Now, L(R) is closely connected to the Axiom of Determinacy, AD.13 A dis-cussion of this axiom would lead us too far astray; see [Ka2009] for an historicaldevelopment as well as technical details. I mention this axiom here only becauseit is needed for the similarity condition of the two structures, see below.

Kanamori remarks

increasingly from the early 1970’s onward consequences of ZF +ADwere regarded as what holds in L(R) assuming ADL(R) [i.e. theassumption that AD holds in L(R)]. ([Ka2009], p. 378)

It has become custom to regard L(R) as ‘the natural inner model for AD’ (ibid.).The second structure which is important for the method to choose between

the different versions of V = Ult-L is a technical generalisation of L(R). Firstof all, notice that L(R) is just L(Vω+1).14 By replacing ω by some cardinal λone obtains the model L(Vλ+1). Now let λ be such that there is an elementary

12L(R) is constructed just like L, but rather than starting the construction with ∅ one startswith R instead. Note that A ∈ L(A) but, in general, A /∈ L[A], which vividly shows that, ingeneral, L(A) 6= L[A].

13The Axiom of Determinacy states that ever subset of the reals is determined.14Recall here the von Neuman hierarchy as given above.

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embedding from L(Vλ+1) into itself with critical point below λ.15 As Koellnerremarks, this embedding condition is ‘the strongest large cardinal axiom thatappears in the literature’ ([Ko2013a]). This structure L(Vλ+1) is the secondstructure considered in the method to choose between the different versions ofV = Ult-L. ‘The difficulty in investigating the possibilities for the structuretheory of L(Vλ+1) is that we have not had the proper lenses through whichto view it. The trouble is that [...] the theory of this structure is radicallyunderdetermined’ ([Ko2013a]).

I now turn to the similarities of the structure theories of these two structures.Firstly observe that one can prove in ZFC that in L(Vω+1) under ADL(R) ω1(=ω+) is a measurable cardinal and in L(Vλ+1) under the embedding conditionλ+ is a measurable cardinal. In this case there is an obvious similarity betweenL(Vω+1) and L(Vλ+1), namely that both ω+ and λ+ are measurable in theirrespective structures, which presents L(Vλ+1) in a light that makes it clear thatthis structure is just the generalisation of (the well-known and understood)L(Vω+1).

Now, under the different versions of V = Ult-L it can happen that moresuch similarities of the structure theories of L(Vω+1) (under the assumptionthat AD holds) and L(Vλ+1) (under the embedding condition) are revealed. Ifthis happens for a version of V = Ult-L, then this can be seen as evidence forthis version of the axiom. If however dissimilarities of the structure theoriesof these two structures are revealed under a version of V = Ult-L, then thiscounts as evidence against this version of the axiom. To put this into moremathematical terms: if and only if the structure theories converge in similarityunder a given version of V = Ult-L does this count as evidence for the relevantaxiom.

A negative convergence-result has already been obtained. Woodin showedthat for one of the V = Ult-L axioms the structure theories do not converge insimilarity. This axiom should hence, according to the described methodology,not be considered to hold in the true universe of sets; [Ko2013a].

The search for similarity leads to, as Koellner terms them, a ‘list of definitequestions’, [Ko2013a]. These questions are answerable, ‘independence is not anissue’ (ibid.). The ‘spectre of undecidability’, which resurfaced in the wake ofthe choice-problem between the different versions of V = Ult-L, seems finally tobe tamed. It is not yet banished, as the analysis of the convergence of structuretheories is not yet completed, but we are now in possession of a method thatcould potentially banish the spectre for good.

This leads us to Koellner’s connection between mathematics and thepluralism/non-pluralism debate. Above I wrote ‘potentially banish’ becausethere is the possibility that the structure theories might diverge for all versionsof V = Ult-L. In this case the method to banish the ‘spectre of undecidability’considered in this paper would fail. It is this thought that Koellner uses to con-nect the mathematical results about the axioms V = Ult-L to the philosophical

15An elementary embedding between two models is a truth-preserving function betweenthese models. For an elementary embedding j the critical point of j is the smallest ordinal αsuch that j(α) > α.

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pluralism/non-pluralism debate.According to Koellner, if a version of V = Ult-L is found under which the

structure theories of L(Vω+1) and L(Vλ+1) converge in similarity,

then one will have strong evidence for new axioms settling the un-decided statements (and hence non-pluralism about the universe ofsets); while if the answers [to the question of convergence] oscillate,one will have evidence that these statements are“absolutely unde-cidable” and this will strengthen the case for pluralism. In this waythe questions of “absolute undecidability” and pluralism are givenmathematical traction. ([Ko2013a], closing words)

The idea then is this: if we can find a version of V = Ult-L for which the struc-ture theories converge, then the ‘spectre of undecidability’ would be banished,the ZFC dilemma resolved and the mathematical results obtained from theanalysis of the relevant structure theories would pull us towards non-pluralism.If on the other hand the structure theories diverge, then the ‘spectre of unde-cidability’ cannot be banished in a very strong sense, which pulls us towardspluralism. Koellner has set up a mathematical test and given us an argu-ment that the outcome of this test should guide us in our positioning in thepluralism/non-pluralism debate. I call this Koellner’s convergence argument.

This concludes my presentation of Woodin’s and Koellner’s arguments. Inthe next passage I critically assess Koellner’s convergence argument.

2.2.5 The Convergence Argument

Koellner holds that through his convergence argument he has given the meta-physical pluralism/non-pluralism debate mathematical traction.

This argument is a clear case of mathematical pull, but thus far the meta-physical debate has not been pulled because the results about the relevant struc-ture theories are not yet obtained. This is a story about a promise of mathe-matical pull and it shows a way how mathematicians engage in a metaphysicaldebate in vivo.

3 What did we learn

The V=Ult-L story shows that some mathematicians are actively working onthe connection between mathematics and metaphysics. This means first of allthat some mathematicians are taking the metaphysical debate seriously. Theyrealise that their position in the pluralism/non-pluralism debate influences theproblems set theory is facing; the non-pluralists are looking to settle CH, thepluralists are not. This makes the metaphysical debate important for the math-ematicians.

The set theorists discussed in this paper do not sit idly by and wait for thephilosophers to resolve the pluralism/non-pluralism issue. Woodin and Koellnerare actively participating in this debate, and they are using mathematical means

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to do so. Woodin is trying to strengthen the case for the non-pluralist by re-solving a deep problem the non-pluralists have: the ZFC-dilemma. The ‘spectreof undecidability’ resurfaces in the various versions of the V=Ult-L axioms, butit is tamed in the sense that formally solvable mathematical questions can beasked which could settle which of the axioms should be accepted. At this point,Koellner connects the mathematics used to the metaphysical discussion with hisconvergence argument: convergence of the relevant structure theories convergefor a certain V=Ult-L axiom is an argument for non-pluralism, divergence anargument for pluralism. Koellner proposes here that future mathematical re-sults should influence our position in the pluralism/non-pluralism debate. Thismeans that we can now generate arguments in a metaphysical debate by doingmore mathematics. Mathematics is set up to pull the metaphysical debate.

Importantly, Koellner does not point to some past events in the history ofmathematics for his arguments in the pluralism/non-pluralism debate. By re-lying on future mathematical results he connects mathematics to metaphysicsin such a way that mathematics actively influences metaphysics. Koellner es-tablishes mathematical pull. Through the convergence argument, the mathe-matician has a way to actively search for an answer to a philosophical questionwithout overstepping the boundaries of the mathematical discipline, and thatthis is possible is an important aspect of the connection between mathematicsand metaphysics.

The V=Ult-L story then shows that mathematics can actively influencemetaphysics through mathematical pull. Philosophers should pay close atten-tion to mathematical pull because mathematical pull generates forceful argu-ments for philosophical debates. The point is that when mathematical pull isestablished, mathematical methods are used to which sufficiently many mem-bers of the mathematical community largely agree. In the V=Ult-L case, thepluralists largely agree on Woodin’s methods. This does not mean that theyall agree on all the argumentative steps of his argument. Magidor for exampleagrees with Woodin on the issues of the truth of the large cardinal axioms andthe implication of their consistency through the methods of the inner modelprogramme, but, contrary to Woodin, Magidor does not believe that an innermodel which could accommodate a supercompact cardinal can ever be found;[Mag2012]. That is, Magidor disagrees on one issue but agrees with the reason-ing at large. This reasoning at large is what I am interested in here. Let me callit non-pluralistic reasoning. Because Woodin uses non-pluralistic reasoning andKoellner’s convergence argument is based on Woodin’s work, the convergenceargument carries force for the non-pluralist. If the relevant structure theoriesshould turn out to converge, then the non-pluralist would have an argumentwhich resolves a deep problem she had, the ZFC-dilemma. This is a success forher position. Divergence on the other hand puts pressure on her because meth-ods she has agreed to have led to a conclusion which opposes her non-pluralism.We have learned from Duhem and Quine that complex systems of thought arenot defeated by a single blow, hence it is unreasonable to hold that divergencein structure theories puts an end to the non-pluralistic position. But in case ofdivergence the non-pluralist has suffered a defeat and now owes us a story how

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to resolve the inner tension in her position.While the convergence argument has direct impact on the non-pluralist, it

has indirect impact on the pluralist. The pluralist did not subscribe to themethods Woodin used. For example, Woodin argued that only through thetruth of the large cardinal axioms can his prediction that some theory T isconsistent be accounted for. I had mentioned that Woodin presupposes non-pluralism here, and hence the pluralist may well disagree on this point. Butdisagreement with the methods used does not mean that the argument has notraction for the pluralist. One position’s successes or defeats of can diminish orincrease the support for its rival positions. In case convergence is observed, thoseset theorists which do not have strong inclinations towards either pluralism ornon-pluralism might well turn to non-pluralism, whereas a defeat of the non-pluralistic enterprise shows that the ZFC-dilemma is unsolvable in a strongsense, which might convince those set theorist of pluralism.

We have seen that mathematical pull can have a profound influence on themetaphysical position of mathematicians. This metaphysical position in turninfluences how mathematics is practised. If we are interested in mathematicsas practised, then we need to pay attention to mathematical pull because itinfluences the mathematics of tomorrow.

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