aa r X i v : . [ m a t h . L O ] F e b Boolean-Valued Sets as Arbitrary Objects
Leon Horsten ∗ Universit ¨at KonstanzFebruary 7, 2020
Abstract
This article explores the connection between boolean-valued classmodels of set theory and the theory of arbitrary objects in roughlyKit Fine’s sense of the word. In particular, it explores the hypothesisthat the set theoretic universe as a whole can be seen as an arbitraryentity, which can in turn be taken to consist of arbitrary objects, viz.arbitrary sets.
Contemporary philosophy of physics aims to develop metaphysical in-terpretations of fundamental current physical theories. In philosophy ofquantum mechanics, for instance, researchers articulate metaphysical ac-counts of what the physical world at the micro-level could be like given ourcurrent quantum mechanical theories. In this article, I want to do some-thing similar for set theory. My aim is to articulate a new metaphysicalview of what the set theoretic world could be like given our current settheoretic theories and practices. ∗ Early versions of this article were presented at the conference on
Abstract Objects andCircularity (LMU M ¨unchen, 6 July 2019) and at the conference on
Set Theory: BridgingMathematics and Philosophy , Universit¨at Konstanz, 28 July 2019). Thanks to Giorgio Ven-turi, Hazel Brickhill, Sam Roberts, Joan Bagaria, Carolin Antos, Neil Barton, Chris Scam-bler, and Toby Meadows for invaluable comments on the proposal that is developed inthis article. forcing , which is an incredibly powerful andflexible technique for producing independence results. There are two mainapproaches to forcing. The first approach, originating with [Cohen 1963],[Cohen 1964], is sometimes called the forcing poset approach . The secondapproach is called the boolean-valued approach . It was pioneered by Scottand Solovay (and independently discovered by Vop˘enka), and was firstdescribed in [Scott 1967]. There is a strong sense in which the two ap-proaches are ultimately mathematically equivalent.The boolean-valued approach, at least in its modern incarnation (asdescribed in [Bell 2005]), is centred around the concept of boolean-valuedsets, which are functions into a complete boolean algebra. Boolean-valuedsets have been studied mostly with the aim of proving set theoretic in-dependence results. Here I want to consider structures of boolean-valuedsets from a metaphysical perspective. I will argue that boolean-valued setscan be seen as arbitrary objects in the sense of [Fine 1985] and [Horsten 2019].Indeed, Fine himself suggested that arbitrary object theory might be ap-plicable to the boolean-valued approach to forcing [Fine 1985, p. 45–46],although his suggestion has hitherto not been followed up.I will develop the metaphysical hypothesis that there is a sense in whichthe set theoretic universe itself is also an arbitrary entity. On the view thatI explore, there is only one mathematical universe. But just as the elementsin it, the set theoretic universe as a whole is an arbitrary entity. And just asthe arbitrary sets in the universe can be in different states, the set theoreticuniverse can also be in many mutually incompatible states.In this article the the boolean algebra-approach to forcing is used as atool to express a metaphysical view. Given the mathematical equivalenceof the boolean algebra-approach and the partial order-approach (“poset-approach”) to set forcing, the metaphysical view that I want to explore canalso be expressed using the partial order-approach, but I will not do sohere. For class forcing , the two approaches are not mathematically equiva-lent. Also from a philosophical view the situation becomes more compli-cated if we take proper classes ontologically seriously. In this article, I willleave these matters aside.The proposal that is explored in this article is tentative, and is deliber-ately kept “open” at several junctures. That is, there are multiple ways in See [Antos et al, forthc]. ∨ , ∧ ,and c , respectively, and I denote the top and bottom elements of an alge-bra as 1 and 0, respectively. An arbitrary F is an abstract object that can be in a state of being some orother F . We may say that an arbitrary F coincides with some F in a state, ortakes a certain value in some state. So, mathematically, an arbitrary F canbe modelled as a function f : Ω → F ,where Ω is a state space , and F is a collection of objects. In order to developa basic feeling for what arbitrary objects are like, let us briefly consider afew simple examples. Example 1
Consider the man on the Clapham omnibus . Such an arbitraryobject could (in some sense) be me, or it could be my next door neighbour. Butthis arbitrary object is neither numerically identical with me, nor with my nextdoor neighbour.
Example 2
Consider an arbitrary natural number . Such an arbitrary objectcan be in the state of being the number 3, but it can also be in the state of beingthe number 4.
Typically, for a property F , there are more than one arbitrary F ’s. For Fine holds that for every F , there is ultimately is no more than one “independent”arbitrary F [Fine 1983, p. 69]. I will not make this assumption here. a strictly between 3 and 6.Then there is also another arbitrary natural number strictly between 3 and6, call it a , which in every state differs from a . So, for instance, in a statewhere a takes the value 4, a takes the value 5. This shows that arbitrary F ’s can be correlated with each other. It has been argued, a.o. by Frege, that there are no arbitrary objects,and this seems still to be the prevailing view. But in the spirit of [Fine 1985]and [Horsten 2019], I will take arbitrary objects ontologically seriously.The aim of this article is not to argue for this metaphysical stance.In many cases, the function range of an arbitrary F , when regardedas a function, consists of specific objects. For instance, in a state where a coincides with the number 4, it takes a specific value. But there are alsoarbitrary objects that can be in a state of being this or that arbitrary object.For instance, an arbitrary arbitrary natural number strictly between 3 and6 can be in a state of being the arbitrary number a , but it can also being ina state of being the arbitrary number a . Such higher-order arbitrarinesswill play an important role in what follows.I will also be liberal in not just considering maximally specific statedescriptions (also known as possible worlds ). Instead, I will also permit asstates situations that are less than fully specified: call them partial states .These partial states can then be modelled as sets of possible worlds. In early work on boolean-valued models, random variables play an impor-tant role. In particular, this is so in the first exposition of the method ofboolean-valued models, Scott’s beautiful article
A proof of the independenceof the continuum hypothesis [Scott 1967].Scott starts his construction of boolean-valued models with a probabil-ity triple h Ω , A , P i , where Ω is a state space, A is a σ -algebra on Ω , and Pis a probability function defined on A . This probability triple is the back-ground of the notion of a random real over Ω , where a random real over Ω is a function ξ : Ω → R Note that this means that it is strictly speaking wrong to speak of the man on theClapham omnibus. See [Frege 1904]. Let R be the collection ofrandom reals. It is easy to see that R is canonically embedded in R (byconstant functions).Scott’s aim is, roughly, to construct a boolean-valued analogue of theclassical rank V ω + , which is the level of the iterative hierarchy where thecontinuum hypothesis (CH) is decided. In this boolean-valued model theaxioms of set theory insofar as they describe V ω + , turn out to be true,whereas CH is false in this model.The language in which Scott describes the initial transfinite levels of theiterative hierarchy has a type-theoretic flavour. In particular, it containsvariables ranging over real numbers, and variables ranging over functionson the reals. The set of natural numbers N is defined in this language as aspecial collection of reals [Scott 1967, p. 95].In the resulting model S , the real number variables range over randomreals (as defined above). The function variables range over a set R R offunctions from R to R that meet an extensionality condition. In the boolean-valued model S , sentences of the language take valuesin a complete boolean algebra B , which is obtained from the boolean σ -algebra A by identifying events that differ from each other only by a set ofprobability 0 (as measured by the probability function P): B = A / ( P = ) .Moreover, B can be seen to have the countable chain condition, which en-tails that B is complete .Then Scott chooses Ω in such a way that S contains many random realsthat are “orthogonal” to each other. This ensures that S | = ¬ CH, where | = is the boolean-valued truth relation. In particular, the “degree” to which tworandom reals ξ and η coincide according to S is “measured” by a booleanvalue, i.e., an element of B . And such an element of B can roughly be takento be the set of states on which ξ and η coincide. In particular, it is required that for each r ∈ R : { ω ∈ Ω : ξ ( ω ) ≤ r } is measurable. But this is not essential for his argument, as Scott himself observes. See [Scott 1967, p. 102]. I.e., up to P = S only verifies the usual set theoretic axioms as far as V ω + goes.But Scott sketches how S can fairly routinely be extended to a boolean-valued model of ZFC that still makes CH false.As objects that take values in states, Scott’s random reals are arbitraryobjects in the sense of [Fine 1985] and [Horsten 2019] (or at least they aremodelled in the same way). But the values of function variables are not nat-ural modellings of arbitrary objects. Going up the hierarchy, functionals,etcetera, are also not arbitrary objects. This “non-uniformity” is eliminatedin later versions of boolean-valued model theory, such as [Bell 2005], as wewill see shortly.The take-away message is that arbitrary objects have played a role inboolean-valued models from the start. Random variables in Scott’s sensehave mostly disappeared from modern treatments of boolean-valued mod-els, and Scott himself already observed that his method for proving theindependence of the continuum hypothesis does not really require them[Scott 1967, p. 110]. Nevertheless, I will argue that in more recent versionsof boolean-valued model theory, arbitrary objects play an even more per-vasive role, albeit in a somewhat less obvious way. Let us now turn to the contemporary approach to boolean-valued models,as described in [Bell 2005].A boolean-valued class model V ( B ) consists of functions u : V ( B ) → B ,where B is a complete Boolean algebra. dom ( u ) can be seen as the quasi-elements of u . And the elements of dom ( u ) are themselves boolean-valuedsets [Bell 2005, p. 21]. This is reflected in the recursive build-up of theuniverse V ( B ) of boolean-valued sets.Given the Stone representation theorem, the boolean algebra B can beconceived of as a field of sets. Each element of B can then be seen asa set of possible worlds , i.e., a partial (or total) state. In other words, B is See in particular [Horsten 2019, chapter 10]. But not entirely: see for instance [Kraj´ıˇcek 2011]. Uniquely so if B is atomic.
6n algebra of states, where the join operation expresses union of states (‘ a or b ’). If in B we have a < b , then the state a is a refinement of state b .The algebra B need not be atomic: there may be no maximally specificstates (‘state descriptions’ in the Carnapian sense). Partitions of unity (asparticular anti-chains in B ) are then especially significant as collections ofmutually exclusive and jointly exhaustive sets of states. In the absence ofatomicity, partitions of unity are the the closest counterparts to the set ofall Carnapian possible worlds.Actuality plays no role in the picture. Just as it makes no sense to askwhich state the fair coin (an arbitrary object!) is actually in (heads or tails),there is no state that V ( B ) is actually in. There are just many states that V ( B ) can be in. Certainly the maximally unspecific top element 1 ∈ B should notbe seen as the actual world. If there are no atoms in B , then there is noteven a candidate of being the actual world in the Carnapian sense.Let us consider identity and elementhood in some more detail. Wedefine [Bell 2005, p. 23, 1.15]: [[ u ∈ v ]] B ≡ _ y ∈ dom ( v ) ( v ( y ) ∧ [[ y = u ]] B ) . (1)This is what it means for some boolean-valued set u to be to some extent amember of the boolean-valued set v . The extent is measured by a booleanvalue.Given extensionality, identity and elementhood are intertwined in settheory: identity also constitutively depends on elementhood. So in theboolean-valued framework we have [Bell 2005, p. 23, 1.16]: [[ u = v ]] B ≡ ^ y ∈ dom ( v ) ( v ( y ) ⇒ [[ y ∈ u ]] B ) ∧ ^ y ∈ dom ( u ) ( u ( y ) ⇒ [[ y ∈ v ]] B ) , (2)where x ⇒ y is an abbreviation of x c ∨ y (with c being the complementa-tion operation of B ).The boolean-valued truth conditions of non-atomic statements are ex-actly what you would expect [Bell 2005, p. 22], so there is no need to con-sider spell them out here. This determines a notion of boolean-valuedtruth. From now on I will write V ( B ) | = φ for [[ φ ]] B = all elements of V ( B ) can straightforwardlybe seen as arbitrary objects. Thus the non-uniformity of Scott’s model,where the range of the first-order quantifiers is somehow distinguished, iseliminated.We have seen that B can be seen as a state space. But given that boolean-valued sets are functions u : V ( B ) → B , boolean-valued sets are arrowsthat “point in the wrong direction” for being arbitrary objects. Their do-main, rather than their range, should be a state space.But we will see that this problem can easily be remedied. First I explainhow V ( B ) itself can be seen as an arbitrary entity. Then I describe how theelements of V ( B ) can also be seen as arbitrary objects.Let us call a ‘partial’ state a situation . Then a maximal anti-chain is acollection of mutually exclusive but jointly exhaustive situations. Everysituation a can, as we will see shortly, itself be seen as a boolean-valueduniverse V ( B ) a . In particular, V ( B ) a will make all the principles of ZFC true.So if { a , . . . , a k , . . . } is a maximal anti-chain (partition of unity) in V ( B ) ,then V ( B ) can be in the state of being V ( B ) a k . More precisely: in the situation a k , the set theoretic universe is in the state of being V ( B ) a k .Consider a boolean-valued set u ∈ V ( B ) . Then for each situation a k , u “is” a boolean-valued set u a k at a k , where u a k is defined as follows: • For all s ∈ dom ( u ) with u ( s ) ∧ a k = u a k ( s ) ≡ a k ∧ u ( s ) ; • For all s ∈ dom ( u ) with u ( s ) ∧ a k = u a k ( s ) ↾ .Now we can say exactly what V ( B ) a is, for any a ∈ B : it is the boolean-valued class model generated by the boolean algebra B a that consists of allelements of the form y ∧ a , with y ∈ B , and which is such that for x , y ∈ B a , x ⋆ y is the same as x ⋆ y in B for ⋆ ∈ {∧ , ∨} , and x c in B a is x c ∧ a in B .Then if B is a complete boolean algebra, so is the restricted algebra B a .If B has the countable chain condition, then B a has it also. And observethat for all a ∈ B , V ( B ) a | = ZFC . On the other hand, the structure V ( B ) might be such that neither CH nor ¬ CH is true in it, but that it could be in8 state where CH is true and it could be in a state where CH is false. Sucha V ( B ) could function as a toy model of a set theoretic universe in whichneither CH nor ¬ CH is true.Let us now turn to the problem of “reversing the arrows”.
Definition 1
For any u ∈ V ( B ) , u ∗ is the function such u ∗ ( a ) = u a for everya ∈ B. Proposition 1
1. u ∗ is uniquely determined by u;2. u is uniquely determined by u ∗ . Proof.
Clause 1 follows from definition 1.Clause 2. of the proposition follows because u = u (where is, as before, the topelement of B). So whether we take V ( B ) to consists of boolean-valued sets u or theircounterparts u ∗ makes no mathematical difference. But the u ∗ ’s are func-tions from the state space B to V ( B ) . Therefore they are arbitrary objects ,or, to be philosophically more correct, they are natural representations ofarbitrary objects. Thus we can regard every u ∈ V ( B ) as an arbitrary objectin the sense of section 2. More in particular: • the u ∗ ’s are total arbitrary objects (since dom ( u ∗ ) = B ); • the state space B of the u ∗ ’s consists mostly of partial states (sincetypically the algebra B will be non-atomic).In category theoretic terms, what we are doing is treating Boolean-valued sets as variable sets . Given a boolean algebra B , one can considerthe category Shv ( B ) of sheaves over B . Then it can be shown that Shv ( B ) and V ( B ) are equivalent as categories [Bell 2005, p. 180]. Thus both V ( B ) and boolean-valued sets u in V ( B ) are arbitrary objectsin the following sense. If the boolean algebra B is atomless, then as we “go Thanks to Toby Meadows for drawing my attention to this. The connection betweensheaf theory and arbitrary object theory was already observed by Fine: see [Fine 1985,p. 47]. B , the universe V ( B ) takes a more specific state V ( B ) a , without everreaching a maximally specific state. Likewise, as we go down B , a typicalboolean-valued set u takes a more specific state u a , without ever reachinga maximally specific state. We have seen how boolean-valued universes, and the sets that they con-tain, can be seen as arbitrary objects. Now I will argue that the set theoreticuniverse as a whole can itself be seen as an arbitrary entity. The slogan is,roughly:The set theoretic universe is the arbitrary V ( B ) .Let us designate the arbitrary V ( B ) as B . The value-range of this arbitraryentity B will range over V ( B ) ’s where B is an element of a large collectionof complete Boolean algebras.Like all slogans, this one has to be taken with a grain of salt. The thesisis not that the set theoretic universe is an entity that can be in the state of being this or that V ( B ) . After all, just as it is unreasonable to hold that thenumber 19 is some pure set or other, so it is unreasonable to maintain thatthe set theoretic universe can be in a state of being some V ( B ) . The point israther that it can be in states that are structurally like, or can be fruitfullymodelled as, V ( B ) ’s. It is sometimes argued that there are different, equally valid conceptsof set, and that it is somehow indeterminate which of these notions is de-scribed in set theory. The position that I am putting forward here is notintended as an articulation of this view. The thought that I am trying todevelop is not that two states of the set theoretic universe describe differ-ent set concepts. Rather, the view is that there is one conception of set thatthe set theoretic universe answers to: a notion of set as an arbitrary object.The central component of the proposed view consists of truth defini-tions for the formulas of the language of set theory ( L ZFC ). I have sketched You might ask: can we not “complete” the value range of B to a complete booleanalgebra B and take the set theoretic universe to be (structurally like) V ( B ) ? But this doesnot work. As is pointed out for instance in [Antos et al, forthc], B is a hyperclass, and V ( B ) therefore does not make ZFC true. B , can also be given: B | = ϕ = : for all B in the value range of B : V ( B ) | = ϕ .A Tarskian truth definition for L ZFC cannot be given in L ZFC ; but we cangive a Tarskian truth theory for L ZFC in an extended language. Similarly,we can give a boolean-valued truth theory for L ZFC in the stronger meta-language L + ZFC , which consists of L ZFC plus a primitive satisfaction pred-icate. And we can of course also in an extended language express thedefinition of truth in B we have given above. In the boolean-valued truthdefinition, boolean-valued sets are assigned to variables. Moreover, theseboolean-valued sets are (in the truth definition) applied to other boolean-valued sets, boolean operations are applied to values of boolean-valuedsets, and so on. But every boolean-valued set can be seen as an arbitraryobject . So we could re-write the boolean-valued truth definition in termsof arbitrary objects instead of boolean-valued sets—although I will not doso here. Thus, ultimately, the view that is suggested presupposes an inde-pendent grasp of arbitrary objects, states, and values of arbitrary objects.In this sense, the boolean-valued truth theory for L ZFC is intended to beinterpreted in arbitrary object theory , which is a part of metaphysics. Theidea is that L + ZFC is itself interpreted as being about arbitrary entities.Since the view that is proposed is meant to be a foundational interpre-tation, it must be autonomous . It must stand on its own two legs: it mustnot be parasitic on any other foundational interpretation.At this point you might worry that the autonomy requirement is notsatisfied: the boolean-valued sets (and hence also the arbitrary objects) towhich the account appeals are defined (recursively) in terms of quantifi-cation over V (over the “ {
0, 1 } -valued sets”), not in terms of quantifica-tion over V ( B ) . But this concern is misguided. The boolean-valued truthdefinition is given in a language ( L + ZFC ), the quantifiers of which range overarbitrary sets . Of course, in order to spell out the truth definition for L + ZFC ,we would have to move to an even richer metalanguage. And, of course,the {
0, 1 } -valued sets are, on the proposed view, special cases of boolean-valued sets: the “traditional” universe V is canonically represented in V ( B ) [Bell 2005, p. 30]. So the view is metaphysically self-contained. In the sense of [Linnebo & Pettigrew 2011].
11n the proposed view, B is the “ultimate” set theoretic universe. Inthis sense, an absolutist interpretation of set theory is proposed. Neverthe-less, there is an obvious connection with multiverse views such as that of[Hamkins 2012], [Steel 2014], [V¨a¨an¨anen 2014]. We have seen how everystate that B can be in determines a boolean-valued set theoretic universe V ( B ) . Moreover, if we take an anti-chain A in B , then every a ∈ A de-termines a set-theoretic universe. In this sense, the set theoretic universecontains many ‘multiverses’. So the position under consideration can belabeled kaleidoscopic absolutism. As a foundational mathematical theory, set theory must be sufficientlyrich to carry out all of accepted mathematics, albeit sometimes in an ex-ceedingly cumbersome way. Thus, in a naturalistic spirit, I take it as a conditio sine qua non that the set-theoretic universe makes
ZFC true, andwe have seen that B does this.As mentioned before, there is a 2-valued universe V that is canonicallyembedded in every V ( B ) . But the idea is that our mathematical experi-ence suggests that the set theoretic world is not such a 2-valued structure[Hamkins 2012, p. 418]:[The] abundance of set-theoretic possibilities poses a seriousdifficulty for the universe view, for if one holds that there is asingle absolute background concept of set, then one must ex-plain or explain away as imaginary all of the alternative uni-verses that set theorists seem to have constructed. This seems adifficult task, for we have a robust experience in those worlds,and they appear fully set theoretic to us.Hamkins takes the independence phenomena to be evidence for his mul-tiverse view; I take them to be evidence for the kaleidoscopic absolutistview.Nevertheless, on the proposed view, the completely determinate andcanonically embedded structure V clearly is of theoretical interest. Forinstance, we of course have V | = CH or V | = ¬ CH , but we don’t knowwhich.One might wonder whether it is reasonable to expect V to be a state of(some, or even every) V ( B ) . If it is, then B will have at least one atom a , The multiverses in V ( B ) (determined by anti-chains) can be turned into multiverses ofclassical, two-valued universes by well-known ultrafilter techniques [Bell 2005, chapter4]. V = V ( B ) a . So then V ( B ) , and therefore also B , will contain at lest onemaximally specific state, i.e., a possible world in the Carnapian sense ofthe word. There are, however, reasons for believing that V is not a state ofa V ( B ) . If for some a ∈ B , V = V ( B ) a , then there is at least one completelyclassical state that the set theoretic universe can be in. Moreover, this stateis then also the only fully determinate state that the set theoretic universecan be in. Set theoretic experience provides no reason to think that thereis any such super-special universe that the set theoretic universe can be.It is then still the case that in L + ZFC we can define the classical class model V in a particularly simple way. But from a foundational perspective, thisdefined class has no special significance.The general picture is as follows. Set theoretic experience (forcing,large cardinal axioms, infinitary combinatorics,. . .) suggests that there are many states that the set theoretic universe can be in. So B has to be suchthat it can be in all and only those states. And this imposes restrictions onwhat V is like and what B is like.Since we want universes with large cardinals to be possibilities, weprobably do not want V to be G ¨odel’s constructive universe L. Indeed,perhaps it can be argued that V contains (many) large cardinals. We haveseen that any B should probably not the {
0, 1 } -algebra. I have also arguedthat B is non-atomic. Beyond this, matters are less clear. In boolean-valued models we can have [[ ξ = η ]] = a for some booleanvalue 0 < a <
1. So it seems that we are committing ourselves to identitybeing to some degree an indeterminate relation.Evans held that indeterminacy of identity is incoherent [Evans 1978].His argument is a simple reductio based on Leibniz’ principle of the indis-cernibility of identicals. Consider any ξ and η that are not determinatelyidentical. Then ξ has a property, viz. being identical to ξ , that η does nothave. So ξ and η are determinately different from each other.It has been observed that, strictly speaking, Evans’ argument does notgo through. In Evans’ argument, Leibniz’ principle is applied to the pred-icate λ z [ z is identical with ξ ] . But then we can only conclude that ξ and η are not identical, not that they are determinately non-identical.13owever, Williamson ([Williamson 2005, section 8]) has shown how anEvans-like argument nonetheless can be carried through with the use oftwo further plausible principles. First, the following inference rule seemsvalid:“From a proof of φ → ψ , infer that if it is determinately the casethat φ , then it is determinately the case that ψ .”Moreover, if it is determinately the case that φ , then ψ . Using these proofprinciples, Evans’ argument can validly be strengthened to conclude thatthere can be no ξ , η , such that (1) it is determinately the case that they arenot determinately identical and (2) it is also determinately the case thatthey are not determinately different.The moral that is often taken from arguments such as these is that thereis no ontological vagueness but only semantic vagueness. That is, I sur-mise, also the attitude that set theorists habitually take, and it is perhapsthe main reason why the V ( B ) ’s other than V are not taken to be candi-dates for being the ‘real’ mathematical universe. Indeed, in the forcingposet approach, the vagueness involved is pretty much officially regardedas semantic in nature, for its counterparts of the ‘ontologically vague’ setsin the boolean-valued approach are the P -names [Kunen 1980, chapter 7, § ξ , η of boolean-valued sets that are notnumerically identical to each other but that are “judged” to be identical bycertain boolean-valued models. Here is a simple example: Example 3
Consider the simple boolean algebra B = { a , b , 1 } with < a , b < and a ⊥ b. Let ξ = { ∅ → ( ∅ → ) → } , meaning that dom ( ξ ) = { ∅ , ∅ → } and ξ ( ∅ ) = ξ ( ∅ → ) = Moreover, let η = { u → v → } , with u and v being the following “anti-correlated” sets: • u = { ∅ → a , ( ∅ → ) → b } ; • v = { ∅ → b , ( ∅ → ) → a } .Then clearly we have, in the strict sense, ξ = η . Nonetheless, a routine buttedious calculation shows that V ( B ) | = ξ = η .
14n a boolean-valued class model, the identity symbol ‘=’ expresses acongruence relation other than the real identity relation: it ‘measures’ thestates in which its arguments coincide . On the proposal under consider-ation, some boolean-valued class model is a good interpretation of thelanguage of set theory. Therefore the proposed view is committed to theclaim that in set theory, the symbol ‘=’ does not express the real identityrelation. As a consequence, it is not threatened by Evans’ argument, norby Williamson’s modification of it.It is at the metaphysical level, i.e., in arbitrary object theory, that wetruthfully say that ξ = η ; in set theory, we truthfully say that ξ = η .So in these two contexts we do not use the identity symbol with the samemeaning. From the debate about mathematical structuralism we are famil-iar with the claim that in many areas of mathematics the identity symbolis commonly not used to express the metaphysical relation of identity—remember the slogan“identity is isomorphism”. Set theory, as a founda-tional discipline, is often taken to be an exception to this phenomenon.On the view that is explored in the present article, this is not correct: theidentity symbol in set theory also expresses a relation that is different from“real” identity. On the view that I have sketched, there is a mathematical universe . More-over, as an arbitrary entity, the universe is an abstract entity . To conclude, itseems natural to say that on the proposed conception the universe is mind-independent . The combination of these three commitments makes the po-sition under consideration a form of mathematical platonism . At the sametime, this position rejects a strong form of truth value realism accordingto which every set-theoretic statement has exactly one of the traditionaltruth values (true, false). However, it is now fairly generally recognisedthat mathematical platonism per se is not committed to this extra thesis,even though most traditional forms of mathematical platonism do sign upto it.Versions of set theoretical platonism without truth version realism havebeen proposed in the literature. In this article, I have suggested one par-ticular such view that takes the set theoretic universe and the sets in it tobe arbitrary objects. 15 do not claim that the view that I have proposed is the only way inwhich forcing models can metaphysically be related to arbitrary object the-ory. I will close by outlining the contours of an alternative way of seeingelements of forcing models as arbitrary objects.The construction of a forcing model is sometimes seen as analogous tothe process of adjoining an object to an algebraic structure. For definite-ness, consider the construction of the ring of polynomials in one variableover R . In terms of arbitrary object theory, this process can roughly beseen as follows. We start with an arbitrary object X with value range R .Then we consider all arbitrary objects that depend on X in the sense of be-ing polynomially determined by X . The resulting collection of arbitraryobjects form a ring.Similarly, given a poset P in a model of set theory M , a generic filter G can be taken to be an arbitrary subset of P . This is the view of Venturi ,who motivates it as follows [Venturi forthc, p. 2]:Intuitively a set is generic, with respect to a model M and aposet P , if it meets all requirements to be a subset of P fromthe perspective of M and nothing more. The elements of P ,called conditions , represent partial pieces of information thatwill eventually give the full description of the generic G . More-over, the dense sets that belong to M represent the proper-ties that a subset of P should eventually have, as consideredfrom the perspective of M . For this reason, a generic set doesnot have a characteristic property that distinguishes it from allother elements of M .In terms of this arbitrary subset G of P , a model M [ G ] can then be seen asa collection of dependent arbitrary objects.This way of connecting forcing models with arbitrary object theorymakes use of a distinction between dependent and independent arbitrary ob-jects. Such a distinction does not figure in the theory of arbitrary objectsthat I favour. But it is the cornerstone of Fine’s arbitrary object theory. Sothe alternative account that I have tried to outline in this section is perhapsbest developed fully within the framework of Fine’s theory. A comparison See [Chow 2008, p. 2], for instance. Not fundamentally, anyway; but notions of dependence can be defined in my frame-work: see [Horsten 2019, section 9.4].
16f this alternative account with the view that was the focal point of this ar-ticle is left for another occasion.
References [Antos et al, forthc] Antos, C., Friedman, S., and Gitman, V.
Boolean-valuedclass forcing.
Forthcoming.[Bell 2005] Bell, J.
Set Theory. Boolean-valued models and independence proofs.
Clarendon Press, 2005.[Chow 2008] Chow, T. A beginner’s guide to forcing. arXiv:0712.1320,2008, 16p.[Cohen 1963] Cohen, P. Theindependenceofthecontinuum hypothesisI.Proceedings of the National Academy of Sciences (1963), p. 1143–1148.[Cohen 1964] Cohen, P. TheindependenceofthecontinuumhypothesisII.Proceedings of the National Academy of Sciences (1964), p. 105–110.[Evans 1978] Evans, G. Can there be vague objects? Analysis (1978),p. 208.[Feferman 2011] Feferman, S. Is the Continuum Hypothesis a definitemathematical problem? Part of the Exploring the Frontiers of In-completeness Project. Available on Feferman’s website.[Fine 1983] Fine, K. Adefenceofarbitraryobjects. Proceedings of the Aris-totelian Society, Supplementary Volume (1983), p. 55–77.[Fine 1985] Fine, K. Reasoning with arbitrary objects.
Blackwell, 1985.[Frege 1904] Logicaldefectsinmathematics. In H. Hermes et al. (eds)
Got-tlob Frege. Posthumous writings. (Translated by Peter Long and RobertWhite)
Basil Blackwell, 1979, p. 157–166.[Hamkins 2012] Hamkins, J. Theset-theoreticmultiverse. Review of Sym-bolic Logic (2012), p. 416–449.17Horsten 2019] Horsten, L. The metaphysics and mathematics of arbitrary ob-jects.
Cambridge University Press, 2019.[Kraj´ıˇcek 2011] Kraj´ıˇcek, J.
Forcing with random variables and proof theory.
Cambridge University Press, 2011.[Kunen 1980] Kunen, K.
Set theory. An introduction to independence proofs.
North-Holland, 1980.[Linnebo & Pettigrew 2011] Linnebo, Ø. & Pettigrew, R. Category theoryas an autonomous foundation. Philosophia Mathematica (2011),p. 227–254.[Scott 1967] Scott, D. A proof of the independence of the continuum hy-pothesis. Mathematical Systems Theory (1967), p. 89–111.[Steel 2014] Steel, J. G¨odel’sprogram. In J. Kennedy (ed) Interpreting G¨odel.Critical essays.
Cambridge University Press, 2014, p. 153–179.[Venturi forthc] Venturi, G. Genericityandarbitrariness. Logique et Anal-yse, to appear.[V¨a¨an¨anen 2014] V¨a¨an¨anen, J.
Multiverse theories and absolutely undecidablepropositions. in J. Kennedy (ed)
Interpreting G¨odel. Critical essays.
Cam-bridge University Press, 2014, p. 180–208.[Williamson 2005] Williamson, T. Vagueness in reality. In: M. Loux & D.Zimmerman (eds)