In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic multiverse to be the best framework for mathematical practice. According to UNIFY, an adequate set theory should be foundational, in the sense that it should allow one to represent all the currently accepted mathematical theories. As for MAXIMIZE, this states that any adequate set theory should be as powerful as possible, allowing one to prove as many results and isomorphisms as possible. In a recent paper, Maddy (2017) has argued that this two principle justify ZFC as the best framework for mathematical practice. I argue that, pace Maddy, these two principles justify a multiverse conception of set theory, more precisely, the generic multiverse with a core (GMH)

de Ceglie, Matteo

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A naturalist justiﬁcation of the GenericMultiverse with a corede Ceglie, MatteoParis-Lodron Universität Salzburg, Österreichdecegliematteo@gmail.comApril 20, 2018AbstractIn this paper, I argue that a naturalist approach in philosophy of math-ematics justiﬁes a pluralist conception of set theory. For the pluralist,there is not a Single Universe, but there is rather a Multiverse, com-posed by a plurality of universes generated by various set theories. Inorder to justify a pluralistic approach to sets, I apply the two natu-ralistic principles developed by Penelope Maddy (cfr. Maddy (1997)),UNIFY and MAXIMIZE, and analyze through them the potential ofthe set theoretic multiverse to be the best framework for mathemati-cal practice. According to UNIFY, an adequate set theory should befoundational, in the sense that it should allow one to represent all thecurrently accepted mathematical theories. As for MAXIMIZE, thisstates that any adequate set theory should be as powerful as possible,allowing one to prove as many results and isomorphisms as possible. Ina recent paper, Maddy (2017) has argued that this two principle justifyZFC as the best framework for mathematical practice. I argue that,pace Maddy, these two principles justify a multiverse conception of settheory, more precisely, the generic multiverse with a core (GMH).1 The MultiverseThe concept of multiverse was born following the discovery of the phe-nomenon of independence in set theory: sentences in the language of settheory, such as the Continuum Hypothesis (CH), that turned to be indepen-dent from the axioms of ZFC. In order to prove these independence results,1set theorists make use of models (universes) diﬀerent from the canonical one:the collection of all these models (universes) constitutes the multiverse.The multiverse then consists of all the models that satisfy the axioms ofcertain set theory: a very liberal notion of the multiverse would admit everypossible universe, for example both ZFC and ZF +¬C, while others woulddeﬁne some criterion, e.g. all the universe without Choice (ZF + ¬C,ZF +AD and so on). In addition, these models contain all the relevant informationon sets. Each of these models is a legitimate universe of set theory; there isno Single Universe. According to proponents of the multiverse, this lack ofunity cannot be repaired in any way and set theory is precisely the study ofthese alternative universes, in which the properties of sets can vary greatlyfrom one to another.From a philosophical point of view, we can classify the various types ofmultiverse by their commitment to the ontological existence of the universes.More precisely, we can recognize two main positions:• a realist multiverse, committed to the full existence of the universesthat form it (e.g. Balaguer's fullblooded Platonism, cfr. Balaguer(1995) and Balaguer (1998));• an anti-realist multiverse, that does not commit to the platonic exis-tence of the universes (this is, for instance, the position defended byShelah in Shelah (2002) and Shelah (2003)).Instead of focusing on the existence of the universes, the mathematical clas-siﬁcation is based on how we build a hierarchy of all these universes. So wecan have:• a broad multiverse, where there is no hierarchy at all, and all the uni-verses have the same status among the others (for instance, Hamkins'multiverse is a broad multiverse, cfr. Hamkins (2012));• a generic multiverse, where we diﬀerentiate between universes using astrong logic (an idea owed to Woodin, in Woodin (2011), from now onGMΩ) or supposing the existence of a core (an idea owed to Steel, cfr.Steel (2014), from now on GMH);• a vertical (or horizontal) multiverse, where all the universes are like ona ladder, bigger and bigger (or on the same level, but we add more andmore universes of the same size);• an hyperverse, that is, a multiverse of multiverse (Arrigoni and Fried-man (2013)).2One can of course apply the philosophical classiﬁcation to the mathematicalone, which yields a realist and anti-realist version of all these multiverses.Not all these multiverses are equally appealing. Some are philosophicallyweak; others are mathematically trivial. All things considered, a consen-sus has emerged to the eﬀect that the GMH is to be regarded as the mostpromising option. It is very intuitive, and hence philosophically well moti-vated. And it is mathematically productive: it validates all the theorems ofZFC, and beyond.2 The naturalistic approach in philosophy ofmathematicsNaturalism in philosophy of mathematics is a methodological approach thatrecommends using mathematical rules and arguments to justify mathemat-ical objects (usually axioms) and statements about mathematics. In mostcases, this means that a new axiom (or theory) should be tested on the basisof its actual or potential beneﬁts for mathematical practice. Every single ar-gument about mathematics should have a speciﬁc mathematical goal. To putthis deﬁnition in a more precise fashion, Maddy (1997) and Maddy (2011)proposed the following principles as a method to test the strength of newaxioms (and, subsequently, theories):Principle 1 (UNIFY) The ultimate goal should be a theory where everystructure and every mathematical object can be modelled.This essentially says that an adequate set theory should be foundational,in the sense that it should allow one to model most (better: all) of currentmathematical practice.Principle 2 (MAXIMIZE) The new axioms should be as powerful as pos-sible, providing all possible results and isomorphisms.A larger number of possible isomorphisms means that our framework has astrong capacity to describe mathematical objects and their relationships.Maddy (2017) further reﬁne UNIFY, and suggests that a theory is foun-dational if and only if it provides the following:• Meta-mathematical Corral;• Elucidation;• Shared Standard;3• Risk Assessment.The ﬁrst thing that comes to mind when talking about a foundationaltheory is the possibility to embed all of mathematics in a single theory, thatis, we should be able to prove, in our foundational theory, something generalabout all mathematics (Meta-mathematical Corral).Another important role of the foundational theory is to replace a vaguenotion with a more precise one. For example, consider the notion of con-tinuity which, before Dedekind's work, was somewhat ambiguous. It wasprecise enough to generate the calculus, but not enough to become a tool inthe proof of fundamental theorems. Only after it was deﬁned in set-theoreticterms it became certain and useful for any purpose.A foundational theory T should provide us with a Shared Standard todecide if a proof is actually part of mathematics or not. Put diﬀerently, Tshould be the judge of what counts as a proof in mathematics and whatnot: a formal derivation in the foundational theory should be regarded asthe standard proof in mathematics.Finally, we should be able to use our foundational theory to assess anynew mathematical object that is considered dangerous and suspicious, andto determine how risky it is to use it.Maddy's MAXIMIZE can be rephrased in such a way as to relate it toShared Standard (cfr. Maddy (2017)): the foundational theory should bea Generous Arena in which we can hope to ﬁnd, study and analyze everymathematical object. According to the revised principle, all the possiblemathematical structures can co-exist in the foundational theory, and theirinterrelations can be studied in such a theory. Two observations are in or-der. First, MAXIMIZE* doesn't go beyond Maddy's original principles, it isjust a way of articulating them diﬀerently. Second, it is worth noting thatthe reference to isomorphisms contained in MAXIMIZE is still present inMAXIMIZE*: if all the various structures from mathematics can co-exist ina single framework, the total number of isomorphisms grows.3 A justiﬁcation of the Generic Multiverse witha coreMaddy (2017) has recently argued that, from a naturalist point of view, theonly possible choice as a foundation for mathematics is the Single Universegenerated by ZFC. She concludes:In sum, then, it seems to me that the familiar set-theoretic foun-dations, rough and ready as they are, remain the best tool we have4for the various important foundational jobs we want done.(Maddy(2017, p.53))Her argument mostly relies on UNIFY: the Single Universe still provides abetter foundation than the multiverse, whence there is no need to adopt thelatter. Now, if one considers Hamkins' broad multiverse (Hamkins (2012)),Maddy's assessment is correct. Because the broad multiverse is so liberal asto accept every possible universe (including e.g non-well-founded universes,trivial universes, and so on), the foundational power of the set theoreticframework is lost. In Hamkins' view, the main job of set theory is to dealwith diﬀerent kinds of constructions, which can verify, or falsify, the sameset-theoretic claims (suche as CH). In such a framework, there is no reasonto ban a particular universe: they are all perfectly legitimate model theoreticconstructions, and thus part of the multiverse.However, Hamkins' conception falls short on exactly this: consideringthe naturalistic principle UNIFY and the foundationality features that comewith it (Meta-mathematical corral, Risk assessment, Elucidation and SharedStandard), his multiverse cannot provide all these features. The main reasonis that there is no bridge between universes: every universe is isolatedfrom the others, with its particular concept of set and membership. Themain consequence is that we cannot deﬁne a shared notion of proof, norprove or deﬁne anything shared between all the universes. By contrast, theSingle Universe V delivers all these features: we can prove something generalabout all mathematics in it, since it can all be reduced in one manageablepackage (Meta-mathematical Corral); we can use set-theoretic deﬁnitions toclear otherwise ambiguous mathematical objects (Elucidation); it is a verygood standard for what counts as a mathematical proof (Shared Standard);and, ﬁnally, it is very useful when trying to assess the problems with the useof a certain mathematical object or method. All in all, we can say that ZFCdoes a very good job as a foundation for mathematics.However, Maddy appears to only focus on UNIFY, and to disregard MAX-IMIZE. The main advantage of any pluralistic framework is that it allows forthe existence of many more structures and objects than the Single Universe.For instance, in the usual set theoretic universe V we must chose which kindof set theory is actually generating the universe (for example, the V generatedfrom ZFC is radically diﬀerent from the universe generated by ZF + AD),thus restricting the available objects and structures. By contrast, in themultiverse all these diﬀerent universes would live side by side, ready to bestudied and compared within the same framework.This is perfectly in line with the naturalistic principle MAXIMIZE: a the-ory is preferable over another if it proves more isomorphisms. In other words,5our theory should be a Generous Arena, where we can compare, study andanalyze as many objects and structures as possible. Having all these struc-ture in the same framework is very helpful for mathematical practice: onecan prove general results about them, search for patterns, and so on.But how are we sure that the multiverse can actually prove more iso-morphisms? To see this, let us consider the most simple case: a multiversemade of two very similar universe, that diﬀer for the least possible amount.In particular, let's assume that one of these universes is actually equal toV , and the other diﬀers from it for just one structure. In this case, we canprove that most of the structures of A are isomorphic to the correspondentstructure of A′. We then get new isomorphisms (in this case particular case,trivial ones) that are unprovable in V alone.Returning to the more complex (and not trivial) case of a full multiverse,we have shown that our new framework is a more Generous Arena than theSingle Universe: there are more mathematical structure and objects than canco-exist in it, and one can study their interrelations.If the only possible multiverse was the broad multiverse, Maddy's diagno-sis would still be correct: the advantages of MAXIMIZE don't counterbalancethe disadvantages of the lost foundationality. The naturalist pluralist, then,would need to ﬁnd a multiverse conception that is powerful enough to satisfyMAXIMIZE while retaining the foundationality of UNIFY. I would like toargue that the generic multiverse with a core GMH (Steel (2014)) is preciselywhat the pluralist needs.In this kind of multiverse, we deﬁne a criterion to narrow the possibleuniverses, thus compacting the multiverse in a coherent plurality of uni-verses. This criterion is the existence of a core: indeed, the GMH is builtaround a common core of set theoretic truths, common to all universes of themultiverse. This universes are all extensions of ZFC, or, semantically, allthe universes of the GMH are extensions of V . Such a construction ensuresthe satisfaction of both UNIFY and MAXIMIZE.Indeed, since the GMH is a multiverse, our argument above for MAXI-MIZE still applies: we can prove more isomorphisms in it than in the Sin-gle Universe. But how foundational is the GMH? Recall, Maddy's mainargument against the multiverse was that it wasn't foundational enough.However, the argument no longer applies in the present case because of theexistence of the core. As a matter of fact, since this core is common to alluniverses, we can easily deﬁne in it all the foundational feature considered byMaddy. The core assures us that the rules are the same in every universe, soour deﬁnition of proof will be preserved across the whole multiverse (SharedStandard). Also, the language and rules we are using will be the same inall the multiverse, so a deﬁnition of a mathematical object in one universe6will still be understandable in other universes (Elucidation). Moreover, if amathematical object clashes with something in our core it will be problematicto use it in every universe of the multiverse (Risk Assessment). Finally, wewill be able to prove something general about not only mathematics, but alsothe multiverse itself, since it will be included in the core (Meta-mathematicalCorral). Moreover, this core is ZFC itself, so we are sure that the GMH isas foundational as V : everything that is true in ZFC, is true in the GMH .Thus, we can conclude that the GMH satisﬁes both MAXIMIZE andUNIFY, so, from a naturalistic point of view, we can only conclude, paceMaddy, that is better than the Single Universe as a framework for mathe-matical practice.4 Concluding remarksIn conclusion, I have argued that, from a naturalistic point of view, theGMH is better than the Single Universe as a framework for mathematicalpractice. This is because is just as foundational as the Single Universe, thussatisfying the principle UNIFY, and it proves more isomorphisms than theSingle Universe, thus satisfying MAXIMIZE.ReferencesArrigoni, Tatiana and Sy-David Friedman (2013). The hyperuniverse pro-gram. In: Bulletin of Symbolic Logic 19, pp. 7796.Balaguer, Mark (1995). A Platonist Epistemology. In: Synthèse 103, pp. 303325. (1998). Platonism and Anti-Platonism in Mathematics. Oxford: OxfordUniversity Press.Hamkins, Joel David (2012). The set-theoretic multiverse. In: Review ofSymbolic Logic 5, pp. 416449.Maddy, Penelope (1997). Naturalism in Mathematics. Oxford: Oxford Uni-versity Press. (2011). Defending the Axioms. Oxford: Oxford University Press. (2017). Set-theoretic Foundations. In: Foundations of Mathematics. Ed.by et al. Caicedo Andrés Eduardo.Shelah, Saharon (2002). The future of Set Theory. In: Israel MathematicalConference Proceedings. Ed. by Haim Judah. (2003). Logical Dreams. In: Bulletin of the American Mathematical So-ciety 40, pp. 203228.7Steel, John R. (2014). Gödel's program. In: Interpreting Gödel. Ed. byJuliette Kennedy. Cambridge University Press.Woodin, William Hugh (2011). The Continuum Hypothesis, the Generic-Multiverse of Sets and the Ω Conjecture. In: Set Theory, Arithmeticand Foundation of Mathematics: Theorems, Philosophies. Ed. by JulietteKennedy and Roman Kossak. Cambridge: Cambridge University Press.8

A naturalistic justification of the generic multiverse with a core

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A naturalistic justification of the generic multiverse with a core

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