Angels Needle

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How Many Angels Can Dance on the

Point of a Needle? Transcendental

Theology Meets Modal MetaphysicsJohn HawthorneMagdalen College, Oxford, OX1 4AU, United Kingdom [email protected]

Gabriel UzquianoPembroke College, Oxford, OX1 1DW, United Kingdom

We argue that certain modal questions raise serious problems for a modal meta-physics on which we are permitted to quantify unrestrictedly over all possibilia.In particular, we argue that, on reasonable assumptions, both David Lewis’s modalrealism and Timothy Williamson’s necessitism are saddled with the remarkableconclusion that there is some cardinal number of the form @

such that there

could not be more than @

-many angels in existence. In the last section, we

make use of similar ideas to draw a moral for a recent debate in meta-ontology.

In this paper we aim to shed light on an undeservedly disparagedmetaphysical question, one often used to parody medieval scholastics:How many angels can dance on the point of a needle?1 It is not clearthat many scholastics were guilty of devoting time to this question.There appears to be no discussion of this question in the work of Thomas Aquinas, whose attempt to work out the nature of angels by

pure reason earned him the title of Angelic Doctor. In any case, there isnothing for them to have felt guilty about. We think that there is muchto be learned from a proper assessment of this question. Modern com-mentators have the advantage of the tools provided by contemporary set theory, which was unavailable to scholastic commentators from

1 One source is D’Israeli (1875, p. 18), who writes:

The reader desirous of being merry with Aquinas’s angels may find them in Martinus

Scriblerus, in Ch VII who inquires if angels pass from one extreme to another withoutgoing through the middle ? And if angels know things more clearly in a morning? How

many angels can dance on the point of a very fine needle, without jostling one another?

Some earlier references to this question in the seventeenth century can be found in Sylla

2005.

Mind, Vol. 120 . 477 . January 2011 ß Hawthorne and Uzquiano 2011

doi:10.1093/mind/fzr004 Advance Access publication 13 April 2011

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antiquity. Our results will not merely be of theological interest. It turnsout that coincident angels raise potentially damning problems for someof the most prominent metaphysics of modality.

1. Preliminaries

First, some scene setting. To forestall any suspicion of heresy, weembrace the possible existence (and indeed actual existence) of angels. On location issues, there have been many opinions. We stateour own, though here is not the place to defend them at length.

To begin, we maintain that angels can literally occupy places — in

just the same sense that bodies can. Admittedly, this has been chal-lenged by Boethius and many learned scholars who maintain that ‘in-corporeal things do not exist in a place’.2 Yet we are partly encouragedin our opinion by the Angelic Doctor, who rightly insists that ‘it isbefitting an angel to exist in a place’ (Aquinas, 1948, part I, q. 52, art. 1).

Is occupation of an place by an angel a primitive fact? Or else is itparasitic on the application of angelic power to a place?3 Here we tendtowards the former view, this time in defiance of the Angelic Doctor,who follows the Damascene in the latter opinion, and is led to think that angels do not occupy places in the same sense as bodies can.

We agree with the Angelic Doctor that angels can lack positivedimensive quantity. However, while he holds that while Angels falloutside the genus of dimensive quantity, we hold that they can havezero dimensive quantity, thus embracing a view he dismisses as mani-fest deception, namely that the indivisibility of angels is like that of apoint. Given our conception of the dimensive quantities of angels, weconclude that — perhaps excepting special cases of bilocation — angels

can occupy a single point at any given time. Finally, we assume that itis possible for more than one angel to occupy a single point at thesame time. The Angelic Doctor has denied this on the grounds that anangel is present in its place by its effects and that two things cannot beimmediate causes of one and the same thing. We answer that, first, wedoubt that an angel is in a place by the goings on in that place being itsimmediate effects and, second, that since many events can occur at asingle place, the considerations given do not establish the conclusion.4

2 In De Hebdomadibus , as reported in Aquinas 1948, part I, q. 52, art. 1.

3 See Aquinas 1948, part I, q. 52, art. 1.

4 We note in passing that the dialectic of this paper would be largely unchanged if we held

the view that angels are extended but can be colocated.

54 John Hawthorne and Gabriel Uzquiano




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We shall not primarily be concerned with what it means for an

angel to dance, rather than, say, run. Our focus is on the more central

question of how many angels can occupy a single point — say, one

that forms the boundary point of a perfectly sharp pin or needle — at asingle moment in time. Contrary to certain philosophers of antiquity,we shall assume that it is perfectly possible for there to be infinitely

many beings in existence at the same time — ‘actual’ as well as

‘potential’ infinities are possible.5

We will also be making certain natural mereological assumptionsabout angels. That they can be spatially point-sized does not show that

they can lack proper parts. (After all, a fusion of two cohabiting angels

is spatially point sized and yet has proper parts.) Indeed, we do notwish to assume that angels lack — or can lack — proper parts — atleast one of us is tempted to the hypothesis that God is part of allthings.6 However, we do assume that no one object is simultaneously

a fusion of each of two different pluralities of angels. A fusion

of Michael, Gabriel, and Raphael, for example, is not a fusion of

Michael and Gabriel, as some of its parts, namely, some — even if not all — of Raphael’s parts, do not overlap either Michael or

Gabriel.7 Among other things, our assumption has the consequencethat no angel has another angel as a proper part — a consequence we

applaud. (To see this, suppose, say, that Raphael is part of Michael.

Then, any part of Raphael would thereby be a part of Michael andtherefore overlap him, with the consequence that a fusion of Michael

and Gabriel would also be a fusion of Michael, Gabriel, and Raphael.)Those ‘naturalistic’ philosophers who have no truck with our theo-

logical pursuits will no doubt find ways to apply the ensuing discus-sion to their own narrow concerns. For example, while they may

repudiate angels, they are typically more accommodating to particleshaving integral spin — otherwise known as bosons — in modern par-

ticle physics. Such particles are generally thought to be point-sized.Moreover, according to the spin statistics theorem, while fermions —point-particles with half integer spin — cannot be colocated, bosons

5 Aquinas’s view is more nuanced: he claimed that God represents an actual infinity but

that no other actual infinities exist. See Aquinas 1948, part 1, q. 7, art. 1.

6

See Hudson 2006 for a discussion of a similar hypothesis.7 We will say that an object x is a fusion of the Fs if and only if every F is part of x and for

any y , y is part of x if and only if y overlaps one of the Fs. Our definition of fusion is in line

with Tarski 1956 and Lewis 1991, but not with Simons 1987. See Hovda 2009 for the significance

of this difference.

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are perfectly well able to cohabit a single spacetime point. An analo-

gous modal puzzle hence arises for bosons.8

2. Against offensive arbitrariness

The reflections so far have yielded a partial answer: ‘At least two’. But

let us, ambitiously, press on. The answer ‘Exactly seven’ is, prima facie,

offensive to reason. While it seems perfectly imaginable that there are

in fact at most seven angels on the point of any needle, it would bemost surprising if some particular finite number provided a necessary

upper bound. We would after all, find it ridiculous to be told thatthere are, of necessity, at most seven happy zebras in reality. The

proposed upper bounds seems no less ridiculous: in each case, the

necessity seems an unhappy marriage of the brute and arbitrary. If the true bounds of necessity would appear totally arbitrary to thehuman intellect, then our capacity to reason and theorize about mo-

dality is radically more impoverished than we imagine. We shall not

indulge in pessimistic scepticism here — that is not to be expected of

creatures made in God’s image. We could just about imagine that theanswer ‘exactly seven’ is not brute but has some further explanation.

Perhaps God sees that it is immoral for more than seven angels to

cohabit, and ensures that morally inappriopriate cohabitation — whilerife in the human realm — never occurs in the angelic realm. (If angels

were essentially good and sufficiently knowledgeable, they would seeto this themselves.) We shall assume in what follows that there are no

such surprising constraints flowing from the moral or the aesthetic,

but that they flow from the structures provided by logic, pure math-

ematics, and the abstract metaphysics of concrete being. In particular,we shall assume that no such considerations rule out the possibility of infinitely many concrete beings existing at the same time. And given

that there could be infinitely many concrete beings in existence and

there could be colocated angels, it seems clear that no considerationsfrom abstract metaphysics, logic, or mathematics would militate

against the possibility of infinitely many colocated angels.

8 Note that even if there is an upper bound on boson cohabitation dictated by the laws of

nature, that does not settle the modal question, at least on the standard assumption that the

laws of nature are contingent.





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3. A Cantorian reminder

We have achieved a fuller answer: ‘infinitely many’. However, once

the Cantorian hierarchy of transfinite cardinals is revealed to usthrough the natural light of reason, we realize that this answer isstill only partial.9 The coarse answer ‘infinitely many’ belies a rangeof finer-grained distinctions. After all the familiar finite cardinalscomes @0, which is the first transfinite cardinal, that giving the car-dinality of the set of natural numbers. And after @0 comes @1, which isdefined as the next transfinite cardinal. Unfortunately, we know very little about how @1 compares with the cardinality of other familiar setssuch as, for example, the set of real numbers. We know that the set of

real numbers has size 2@0 but not whether the latter cardinal is strictly greater than — as opposed to exactly — @1.

10 But be that as it may, wedo know that after @1 comes the next transfinite cardinal @2, and weknow further that after each @n , comes @n +1, which is its immediatesuccessor. After all these, comes @

!, which is the least cardinal greater

than every @n for n 2!. And after @!

comes @!+1. As you may expect

by now, after all cardinals of the form @!+n , for n 2!, comes @

!+!.And so on. Quite generally, for every ordinal , there is a cardinal @

.

The Cantorian scale of alephs provide us with a dizzying array of candidate resolutions to our theological quandary.But let us not forget another crucial tenet of the Cantorian frame-

work: every set in the realm of the finite and the transfinite hasan aleph as its cardinal number, but, conversely, every aleph is thecardinal number of a set. More importantly, no aleph comes close tomatching the magnitude of the Absolute, ‘the veritable infinity ’ which:

… cannot in any way be added to or diminished, and it is therefore to belooked upon quantitatively as an absolute maximum. In a certain sense it

transcends the human power of comprehension, and in particular isbeyond mathematical determination. (Cantor, 1932a, p. 405)

9 Here we side with Georg Cantor, who writes:

It is my conviction that the domain of definable quantities is not closed off with the finite

quantities and that the limits of our knowledge may be extended accordingly without this

necessarily doing violence to our nature. I therefore replace the Aristotelian-Scholastic

proposition: infinitum actu non datur with the following: Omnia seu finita seu infinita sunt

et excepto Deu ab intellectu determinari possunt . [All forms whether finite or infinite aredefinite and with the exception of God are capable of being intellectually determined.]

(Cantor, 1932b, p. 176)

10 The assumption that 2@0 ¼ @1 is the Continuum Hypothesis, which is known to be

independent from first-order Zermelo-Fraenkel set theory with choice (ZFC).

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Early on, Cantor went on to conceive of this absolute maximum as anappropriate symbol of the power and transcendence of God:

What surpasses all that is finite and transfinite is no ‘Genus’; it is the singleand completely individual unity in which everything is included, whichincludes the ‘Absolute’ incomprehensible to the human understanding.This is the ‘Actus Durissimus’ which by many is called ‘God.’ (Cantor,1979, p. 290)

But this further thought is of course far from obligatory in subsequentarticulations of the Cantorian framework.

What matters for present purposes is that unless we are prepared toallow for the possibility of so many angels as to match the Absolute,

the scale of alephs exhausts the range of cardinalities we might hopeto assign to them.

4. Indefinite Extensibility

The answer ‘exactly @7 ’ is prima facie offensive to reason. The concernabout arbitrariness for the answer ‘exactly seven’ seem to carry over —with roughly equal force — to the answer ‘exactly @7 ’:

Perhaps a better answer might be to say that while there cannotpossibly be absolutely infinitely many angels in existence, no alephmanages to set an appropriate upper bound on the possible cardinal-ities of angels dancing on the point of a needle:

Indefinite Extensibility : There could not be so many angels as toexceed each and every aleph, but for each , there could be exactly @

-many angels in existence.

This answer does not suffer from the arbitrariness of previous answers,though you may well wonder about the assumption that the Cantoriansequence of alephs exhausts the range of live answers to our question.What might prevent God from creating so many angels on the pointof a needle as to exceed each and every aleph? It is not obvious how, by themselves, the methods of trascendental theology can rule out thepossibility of the existence of absolutely infinitely many angels on thepoint of a needle. We would rather not speculate about the idea that,since the transcendent Absolute is an appropriate symbol for thepower and transcendence of God, it is not to be matched by any other actual infinity.11

11 It is striking that both Aquinas and Cantor reserve the greatest size for the deity.





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5. Plenitude

We have arrived at another live answer to our question: ‘absolutely

infinitely many’. Maybe there could be so many angels on the point of a needle that no aleph in the Cantorian hierarchy could ever do justiceto their quantity. Unlike the offensive answers dismissed earlier, thisstrikes us as a live alternative to Indefinite Extensibility.

But what exactly would it be for some angels to be absolutely in-finite in number? Cantor (1967) used the series of ordinal numbers,which he called , in order to measure absoluteness. In particular, heargued that a collection — or ‘multiplicity ’, to use his term — X isabsolutely infinite and hence not a set if and only if can be injected

into X, that is, there is a one-one map from all the ordinals in thesequence into some members of X.12 Since there is a one-one mapfrom onto the aleph series, Cantor’s criterion gives us that X isan absolutely infinite collection if and only if there is a one-to-onemap from the aleph series into X.

This observation can be used to sharpen the last live answer to ourquestion. To say that there could be absolutely infinitely many angels isto say that there is a one-one map from the alephs into the angels, which

is to say that there could be at least as many angels as there are alephs.Plenitude : There could be at least as many angels as there arealephs.

Plenitude tells us that there could indeed be so many angels as toexceed each and every aleph.

6. Some argumentative tools

How are we to choose between Indefinite Extensibility and Plenitude?And what constraints will our answer place on a proper metaphysicsof modality? Before answering these questions, we need to call thereader’s attention to three potentially important ideas. One of themis a result that is largely beyond dispute. Two are quite natural hypoth-eses about the structure of the universe of set theory. (After sketchingthe consequences of these last two ideas we will entertain an approachto the metaphysics of size that somewhat bypasses them, but which

preserves some of the lessons of the preceding discussion.)12 Cantor relied on implicit assumptions that are far from innocent from a modern per-

spective. His conclusion, however, yields a very elegant criterion for absoluteness. See Hallett

1984, p. 172, for a detailed discussion of Cantor’s argument and his implications.

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6.1 Idea one: a mereological result Call a plurality disperse if and only if there are no two different sub-

pluralities of it such that a single object is a fusion of each of them.13

The plurality consisting of an apple, its left half, and its right half is notdisperse since the apple is a fusion of itself as well as a fusion of four

other subpluralities, namely, the apple and its left half, the apple and

its right half, the apple’s left and right half, and, finally, the apple andboth its left and right halves. In contrast, for example, any plurality of

mereological atoms will be disperse. It follows from our earlier as-sumption that the archangels Michael and Gabriel are a disperse plur-

ality, for the simple reason that each of its three subpluralities results

in a different fusion: a fusion of Michael, a fusion of Gabriel, and afusion of Michael and Gabriel will be different from each other.

Let us say that a fusion is based on a plurality if and only if it is a

fusion of one of its subpluralities. We assume the principle of unre-

stricted composition — that any plurality has a fusion. The result that

we wish to draw attention to is the following:

Remark 1: If a plurality is disperse and more numerous than one ,

then there will be more fusions based on that plurality than there are

members of it .

We draw on the following observation:

Remark 2: If a plurality is more numerous than one , then it has

more subpluralities than members .

Two pluralities have the same size if and only if there is a one-one map

from the first onto the second. A map is a relation which pairs every object in the domain with at most one object in the range of the

relation. Finally, a relation is a one-one map from one plurality onto another if (i) no two objects in the domain are paired with the same

object in the range and (ii) the first plurality is its domain and the

range is exactly the second plurality. (If the range is a proper subplur-ality of the second plurality, we say that the relation is a one-one map

from the first plurality into the second.)Quantification over relations can be simulated by plural quantifi-

cation over ordered pairs. However, we need the domain of individ-

uals to be closed under the formation of ordered pairs. The question is13 As we use the terms ‘plurality’ and ‘subplurality’, to speak of a plurality of certain objects

is to speak of the objects themselves. Likewise, for a subplurality of a given plurality of objects,

by which we mean ‘some of them’.





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how to understand talk of ordered pairs. One option is to takethem to be sets and rely on the closure of the domain under certainset-theoretic operations. Another option is to mimic ordered pairs in

the framework of ‘megethology’: plural logic augmented with mere-ology.14 The appendix of Lewis 1991 shows how to simulate orderedpairs in our framework on the assumption that there are infinitely many mereological atoms. And Hazen (1997) has shown how to relax this assumption further by carrying out the simulation in the contextof an infinite atomless mereology. Either way would suit our dialect-ical purposes, though each option comes with its own costs. Thefirst option requires any two objects to form a set, which, though a

theorem of any Zermelo-Fraenkel-style set theory, makes our resultto depend on a purely set-theoretic assumption. The second optionrequires one to assume fairly robust plural versions of choice as part of the framework of plural logic.15

There is a more serious worry in the vicinity. One way to put thesecond remark is as the claim that if a plurality is more numerous thanone, then there is no one-one map from its members onto its sub-pluralities. But what sense is to be made of a relation between themembers of the plurality and its subpluralities? It is not, after all, as if

an ordered pair can pair an object with a plurality; we can only havean object figure as a second component of an ordered pair. Here wehave to be devious and resort to coding.16 If x is in the domain of a relation R, think of R x as the plurality of objects which the relationR pairs with x . Now think of x as a code in R for the plurality R x .We can now think of a relation R as representing a one-one map fromits domain into some pluralities, namely, those which are coded in R by a member of the domain. So, given a plurality more numerous than

one, our observation becomes the claim that no relation can representa one-one map from its members onto its subpluralities.Here is the (schematic) justification for the claim, which is inspired

by the usual diagonal proof of Cantor’s theorem:

Suppose, for reductio , that there is a relation R that represents a one-onemap from a plurality more numerous than one onto its subpluralities.

14 The term was coined by David Lewis in Lewis 1991.

15

Lewis (1991) identifies some of these principles and their role in the simulation. Any of them fit well with one of the important ideas — limitation of size — we will introduce in a

moment. (This is because limitation of size will entail the existence of a well-ordering of the

universe.)

16 See Shapiro 1991 for discussion of the coding scheme and its limitations.

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Then one such relation — call it a groovy relation — has one memberof the plurality code a subplurality of which it is not itself a member.17

(If a member of the domain, a , codes in R a plurality consisting of a

alone, then let a swap places with another, b , and the result will be a groovy relation.)

Let R be a groovy relation. Now consider the codes in R of subpluralities of which they are not members. Call them the Beatles . Since, by reductio ,R represents a one-one map that is onto, the Beatles must be coded in R by some member of the domain, call him Ringo . Now let us ask the question:‘Is Ringo one of the Beatles?’ Well, if Ringo is one of the Beatles, then wewill deduce that he is not. After all, the Beatles are all and only codes of subpluralities that do not have them as members. If Ringo is one of the

Beatles, then he thereby fails to meet a necessary condition for membershipto the Beatles and we must conclude he is not one of them. But if wesuppose that Ringo is not one of the Beatles, we can prove that he is. Forif he is not, then he is the code of a subplurality that does not contain itas a member and therefore meets a sufficient condition for bona fidemembership to the Beatles.

No groovy relation represents a one-one map from a plurality that is morenumerous than one onto its subpluralities. Therefore, no relation does. Weconclude that if a plurality is more numerous than one, then it has more

subpluralities than it has members.18

We are now one step away from our mereological result. A disperseplurality that is more numerous than one has more subpluralities than

members. But given unrestricted composition, every subplurality will

have a fusion. By disperseness, different subpluralities will have dif-ferent fusions, whence there are more fusions based on the initial

plurality than there are members of it.To the extent to which our result did not depend on any contin-

gencies, it holds necessarily.19

17 This would not be true if the plurality had only one member, for there is only one

relation that represents a one-one map from its only member into its only subplurality, which

is not at all groovy.

18 Although we have given an informal argument, plural comprehension is the only prin-

ciple that is distinctive to the logic of plurals. In this respect, the argument is analogous to the

Cantorian argument for the second-order claim that no binary relation can represent a

one-one map from a concept, say, onto all its subconcepts (see Shapiro, 1991, p. 104).

19 Similar results hold for properties and propositions. Suppose we accept an abstraction

principle that tells us that for every plurality, there is a property of being one of them. Then ananalogous argument will show that for any plurality with more than one member, there are

more identity properties based on that plurality than there are members. Suppose we accept an

abstraction principle for propositions according to which for every plurality of worlds, there is

a proposition that is true if and only if one of those worlds obtains. Then, an analogous





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6.2 Idea two: The Urelement Set AxiomImpure set theory is set theory with urelements, that is, non-sets.In contrast to pure set theory, in which one restricts attention to

pure sets, in the context of impure set theory, many have found itnatural to assume that necessarily, the urelements form a set. Call theproposition that the urelements form a set the Urelement Set Axiom.20

Although independent from the usual axioms of impure set theory (e.g., Zermelo-Fraenkel set theory with urelements plus choice(ZFCU)), it is not uncommon to cite the iterative conception of setby way of motivation. On the iterative conception, sets are built instages of a certain cumulative hierarchy. At stage zero we have the

non-sets. At stage one we form arbitrary sets of urelements — any urelements form a set in stage one. At stage two, we form arbitrary sets built of urelements and or level-one products. At stage omegath,we form arbitrary sets of urelements and products from finite stages.And, more generally, at stage , we build sets from materials thatfigure in prior stages. On this picture, the set of urelements willappear at stage one.21

Yet another consideration in support of the hypothesis has to dowith the alleged universal applicability of mathematics, which is sup-

posed to investigate structures presented by the other sciences. To theextent to which set theory is understood to provide a foundation formathematics, we would expect the universe of sets to be rich enoughto enable us to provide a set-theoretic surrogate for any structurewhatever presented by the other sciences. Without the UrelementSet Axiom, we may not be able to represent certain structures

argument will show that for every plurality of worlds greater than one, there are more prop-

ositions based on those worlds than there are worlds. (An identity property is based on a

plurality of things if and only if, for some subplurality, it is the property of being of theelements of the subplurality; a proposition is based on a plurality of worlds if and only if, for

some subplurality, that proposition is true if and only if one or other of the worlds in the

subplurality obtains.)

20 Call an urelement that has no sets as parts an untainted urelement . A fusion of the empty

set with a trout is undoubtedly an urelement, but not an untainted one as it still has a set as a

proper part. A candidate restriction of the axiom is this: any untainted urelements form a set.

This restricted version of the urelement set axiom would serve our dialectical purposes just as

well.

21 See Lewis 1986, p. 107, and Sider 2009 for support. When we begin with a set of indi-

viduals, the result is a transfinite sequence of stages, which provides us with a map of theset-theoretic universe. If U is the set of individuals:

U 0 = U

U þ1 ¼ U [ PðU Þ

U l =S

<lU , for l a limit

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constituted by non-sets, and this would threaten the universality of mathematics.22

There is a more arcane motivation for the Urelement Set Axiom:

McGee (1997) shows how the axioms of the resulting theory are able tofix the structure of the domain of pure sets in combination withthe size of the universe of all objects.23 However, weaker assumptionswill do when one is willing to avail oneself of plural quantificationover ordered pairs. The idea we introduce in the next subsection —limitation of size — would fit the bill perfectly.

Moreover, the axiom is exactly what you need if you want to avoidrecourse to full second-order logic. However, since we take ourselves

to have enough resources to simulate full second-order quantification,we are not particularly moved by this consideration.While we accord our second idea — the urelement set axiom — a

measure of respect, we would not like to rest too much weight on it.As Nolan (1996, p. 254) emphasizes, nothing of strictly mathematicalvalue would be lost if we were forced to abandon it — in fact, settheorists seem to vary from ignoring urelements as a bothersomedistraction to explicitly assuming, for convenience, that there are nourelements at all. Moreover, Zermelo (1930) sketched a conception of

the universe of set theory with urelements as layered in stages of acumulative hierarchy in which the urelements need not form a set.There is some evidence, then, that the heuristic picture of the universeof set theory as built in stages need not require the assumption thatthe urelements must form a set.

6.3 Idea three: Limitation of size One heuristic thought often used to motivate some of the axioms of

set theory is the limitation of size view on which a plurality forms a setif and only if they are not in one-one correspondence with the entireuniverse of all objects. This principle, which is due to von Neumann(1925), helps motivate some of the standard axioms of Zermelo-Fraenkel set theory (with or without urelements). For example, lim-itation of size yields the axioms of separation and replacement as

22 This consideration has been independently advanced, for example, by Allen Hazen (2004)

and Vann McGee (1997).

23 The result reads as follows: any two models of (schematic) second-order ZFCU +Urelement Set Axiom of the same cardinality have isomorphic pure sets. For a proof and

an account of the difference between schematic and full second-order logic, see the appendix

to McGee 1997. In the context of full second-order logic, our third idea below could be used to

achieve the same purpose.





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immediate consequences. Separation says that any plurality of mem-

bers of a set forms itself a set. Replacement, for its part, says that if a

plurality is in one-one correspondence with the members of a set, then

it does itself form a set. More surprisingly, perhaps, limitation of size yields justifications for the axioms of union and choice.24 As forchoice, von Neumann’s principle delivers the existence of a global

well-ordering of the universe. We know, by the reasoning of the

Burali-Forti paradox, that the ordinals fail to form a set. Therefore,

by limitation of size, we have that the ordinals must be in one-onecorrespondence with the entire universe. However, given that theordinals are themselves well-ordered, a one-one map of the universe

into the ordinals gives us a global well-ordering of the universe, which,in turn, entails weaker formulations of the axiom of choice.25

Against the background of the Cantorian framework, one attractiveconsequence of the von Neumann principle is that it tells us that the

size of the Cantorian aleph series is the only extant size beyond all the

alephs. You may think this is in line with the early Cantorian idea that

the Absolute cannot be increased any further and thus sets a quanti-tative maximum (and furthermore, the limitation of size hypothesis

makes the Cantorian series of alephs an adequate foundation for aperfectly general theory of cardinality — we know from Cantor that if

a plurality is too large to be numbered by an aleph, then the entire

aleph series can be injected into it), but the von Neumann principletells us that this can only happen if there is a one-one correspondence

between the members of the plurality and the members of the aleph

series. So, we can count on the aleph series as an appropriate measure

of its size.The scale of alephs can thus be thought to form a proper founda-

tion for the metaphysics of size by forming a kind of universal ruler, inthe sense that the size of a plurality is determined by its relation to the

ruler. If a plurality corresponds to a notch on the ruler, the aleph atthat notch says what size it is. And if a plurality does not so

24 The claim that limitation of size gives us union is due to Le vy (1968).

25 Not only was von Neumann himself aware of the strength of his axiom, he conceded that

‘one might say that somewhat overshoots the mark’. However, he went on to write ‘I believe I

was not too crassly arbitrary in introducing it, especially since it enlarges rather than restricts

the domain of set theory and nevertheless can hardly become a source of antinomies’ (vonNeumann, 1925, p. 402). We note in passing that one way of motivating the limitation of size

axiom is by the Urelement Set Axiom. A second-order version of the axiom is, in the context

of second-order ZFC, a consequence of the axiom of global choice. The observation can be

extended to the case of second-order ZFCU given the Urelement Set Axiom

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correspond one can deduce what size it is: it matches the plurality of alephs and thus size of the universe. This is in fact the main conse-quence of the von Neumann principle we will be using in what

follows.The von Neumann principle is not offered as a mere contingency.

So let it be part of the third idea that it holds necessarily.

7. Plenitude under fire

If all three ideas are embraced, two distinct arguments againstPlenitude soon become available:

(1) The Argument from the Urelement Set Axiom:Consider, for reductio , a world where there are exactly as many angels as there are actual alephs. And suppose that the domainof pure sets does not vary from world to world. Since, presum-ably, no angel is a set and they are therefore urelements, itfollows, by the Urelement Set Axiom, that there is not only aset of urelements but, by separation, a set of angels. It follows,by Plenitude, that the entire sequence of alephs is in one-one

correspondence with the members of a set. But, by replace-ment, the alephs themselves would have to form a set,which would in turn have an aleph as its cardinality. But theset-theoretic antinomies prohibit such a conclusion. HencePlenitude must fail.

A version of this argument has been used by Sider (2009) in connec-tion to Williamson’s necessitism.

(2

) The Limitation of Size Argument:We assume that a fusion of urelements is itself an urelement.And, since angels are themselves urelements, a fusion of angels must itself be an urelement. Given the mereologicalresult, there must be strictly more urelements than there areangels. After all, our result tells us that there are strictly morefusions of angels than there are angels. Limitation of size tellsus that the size of the urelements is at most the size of thealephs. Let us further make the benign assumption that

the size of the aleph series does not vary from world toworld. Given Plenitude, we are forced to conclude that thesize of the angels matches the size of the actual alephs. Butnow, by our mereological result, we must conclude that there





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are strictly more urelements than there are alephs, which

contradicts limitation of size. We conclude that Plenitudefails again.

8. Modal variation

We have been labouring under the assumption that that the alephsequence, and likewise the domain of pure sets, remains constantfrom world to world. What if we relax that assumption? There aretwo ways to do this. One is to allow for modal variation in the domain

of pure sets and yet to insist on a modal maximum for such a size:there is some world, w M , where the alephs (and the pure sets) havemaximal size and hence there is no possible world where we can findstrictly more of them. Another denies a modal maximum for the sizeof the alephs or that of the pure sets: for any world w 1 there is a worldw 2 where there are strictly more alephs and pure sets than thereare at w 1.

If we choose the first option, then there is a natural analogue of Plenitude: there is a maximum size for the plurality of angels, which is

given by the maximum size for alephs and pure sets. But the analogueview will be undermined by either the Urelement Set Axiom or theLimitation of Size argument. If, instead, we choose the second option,then we no longer have a natural analogue for Plenitude available.Selecting any particular pure set size — the size that the alephs actually have, for example — as the modal maximum for the size of angelscommits us to unwanted arbitrariness. It would be more natural toopt for an analogue of Indefinite Extensibility instead. Just as the size

of the alephs is indefinitely extensible, so is the size of the angels.However, von Neumann’s limitation of size principle is not parti-cularly appealing in the context of Indefinite Extensibility. For sup-pose we grant that there could have been strictly more alephs thanthere are in our world. And let us grant, in addition, that there couldhave been strictly more angels than there are alephs in our world, calltheir size . Whence now the modal guarantee that in any worldin which there are exactly -many angels, there must be strictly more than -many alephs? Such a claim appears to institute a neces-

sary connection between the sizes of distinct existences — alephs andpure sets and angels — that is not at all easy to justify. Having noticedthat there could be exactly -many angels, on the one hand, and thatthere could be exactly -many alephs and pure sets, on the other,

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it becomes mysterious why those states of affairs should not be com-possible. But once this compossibility is allowed, von Neumann’slimitation of size principle fails, since there will be strictly more

fusions of angels at such a world than there are alephs at that world.We shall not be speaking further to the view that the size of the

alephs is modally inconstant. Such a view is certainly a minority view.And in any case it does little to disturb the intellectual thrust of thediscussion so far, which favours the indefinite extensibility vision.

9. Interlude

The materials presented thus far strongly point towards IndefiniteExtensibility. Does this provide us with a stable resolution of ourscholastic inquiry, a secure result within transcendental theology?Matters are not quite so simple. In the remaining discussion weshall pursue two separate themes. First, we shall attempt to show that a number of prominent views in the metaphysics of modality cannot be squared with the Indefinite Extensibility picture. If wehold that picture as a theological fixed point, we are given new andsurprising resources to select certain metaphysics of modality over

others. Second, note that the space of alternatives explored thus farhas been set by the Cantorian theory of cardinality, minimally supple-mented by the assumption that we can make sense of the size of theentire sequence of alephs. However, it is not at all clear that this is theappropriate foundational framework. We sketch an alternative frame-work for thinking about sizes in which certain structural assumptionsendemic to the set theoretic framework are exposed as non-obligatory and with which another important candidate resolution to our theo-

logical quandary presents itself.

10. Trouble for modal realism

Suppose with Lewis (1986) that possible worlds are existing concreteuniverses. Suppose further that we embrace Indefinite Extensibility:there could not be so many angels to exceed each and every aleph, butfor each , there is a world in which there are exactly @

-many angels.

The trouble comes when we open up our quantifiers and describe the

structure of the pluriverse, the posited reality of multiple concreteuniverses.

When we take in the universe in one sweep, some mereologicaldecisions have to be made. Are we to allow fusions that are composed





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from objects from multiple worlds? For example, is there a object

that has David Lewis as one part and a possible talking donkey as

another? Lewis explicitly allows such fusions and, indeed, gives them

substantive work to do in his metaphysics. (For example, in Lewis1986 he recommends the identification of properties of individualswith sets of individuals, where such sets are typically transworld.

However, in Lewis 1991 he identifies sets with fusions of singletons.

Therefore, properties of individuals become fusions of singletons,

where such fusions are generally transworld.) It would be exceedingly strange, moreover, to disallow such composition. If the mereologicalgods are sufficiently liberal as to allow the fusion of David Lewis with

objects that are spatially and/or temporally distant, why should spatio-temporal disconnectedness provide an insuperable barrier? (Note that

such barriers would be particularly surprising if, as he avers in Lewis1991, mereology is analogous to logic. From that perspective, a prohi-

bition on transword composition would be analogous to a bar onconjunction introduction for propositions about different worlds.)

Once transworld composition is allowed, the very sorts of problemsthat afflicted Plenitude re-emerge, even for the proponent of

Indefinite Extensibility. If for any , there are exactly @

-many angels at one world or another, then there will be at least as many

angels across the pluriverse as there are alephs altogether. (Note that

this result does not even require the thesis — which Lewis in any caseembraces — that angels are world-bound, that is, the thesis that angels

at different worlds are numerically distinct.) The cardinality of the

angels across the pluriverse will have to be greater than any aleph,

since by hypothesis, any aleph is surpassed by the angels at somecorner of the pluriverse. The confinement of angels to aleph-sizes

within worlds still leaves us with a pluriversal size that matches thesize of the alephs and the pure sets across the pluriverse. With this

result in place, both arguments against Plenitude can now be brought

to bear against any such description of the pluriverse. The UrelementSet Axiom would force the angels across the pluriverse to form a set,which cannot be allowed for the reasons given above.26 And the com-

bination of the von Neumann principle and the mereological result

would entail that the angels across the pluriverse both do and do not

match their fusions in size.

26 A similar argument given in Nolan 1996, pp. 246–7, and adapted to the constant domain

metaphysics discussed below by Sider (2009).

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It thus seems that if one is a modal realist, one can embrace neither

Plenitude nor Indefinite Extensibility: one has to claim that for some, @

does in fact provide a modal upper bound on the size of the

angels can have in a world. One could say that at some world this sizeis attained by the angels. Or one could instead says that while it isnever attained, any lower cardinal is attainable. Either way, the positedupper bound reeks of the very sort of arbitrariness we have been atpains to avoid.

To ward off the Forrest-Armstrong argument against his favouriteprinciple of recombination, Lewis has elsewhere noted that his brandof modal realism accepts certain size and shape limits on recombina-

tion.27

But notice that a restriction on the possible size and shapesof concrete universes do not help us much in a context where angels

can be packed into a single point. While modal restrictions onsize and shape of universes may not seem too outrageous, the restric-tion envisaged here — to a particular aleph upper bound on the sizeof angels — cannot be swallowed so easily. In so far as we know Indefinite Extensibility to hold, we also know that modal realism isfalse.

In a slightly different but related context, Nolan (1996) has recom-mended that Lewis drop the Urelement Set Axiom. Note that inthe present context such an adjustment is not sufficient to solve the

problem. After all, the proponent of Indefinite Extensibility whois also a modal realist has two arguments to deal with, not one.Even with the Urelement set axiom dropped, a von Neumann limita-

tion of size principle plus our mereological result will undo IndefiniteExtensibility.

11. Trouble for necessitism

In Williamson 1998 and Williamson 2002, Timothy Williamson has

argued for a view according to which the very same objects exist at allworlds. So, for example, David Lewis necessarily exists. The appear-ance of contingency is explained away via the hypothesis that Lewis isconcrete at some worlds though failing to be concrete at others,together with the speculation that ‘exists’ in natural language some-times means ‘concrete existence’ and not the existence simpliciter thatconcerns logic and metaphysics.

27 See Lewis’s discussion of recombination in Lewis 1986, pp. 102–4.





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On this view, then, every possible angel exists. Of course, a possibleangel need not be an angel — perhaps one has to be concrete in orderto be angelic. (We leave aside the issue of what the concrete/non-

concrete distinction might come to if, as the Angelic Doctor seemedto hold, an angel is form without matter.) Yet, even if we take care todistinguish bona fide angels from bona fide entities that are merely possible angels, the seemingly happy solution provided by IndefiniteExtensibility becomes destabilized. After all, if for any , there couldbe exactly @

-many angels, then there actually are at least as many

possible angels as there are alephs.Now while many possible angels may not be angels, it seems clear

that a possible angel is not a set. (For one reason, it seems that a set isnecessarily a set and that nothing could be both a set and an angel.)Then, as Ted Sider has pointed out (in Sider 2009), the Urelement SetAxiom will make trouble for Williamson’s position. Moreover, even if that axiom is dispensed with — in accord with Nolan’s advice — oursecond argument will proceed along familiar lines. Assuming unrest-ricted composition, there will be strictly more fusions of possibleangels than there are possible angels. But once von Neumann’s prin-ciple is assumed, this cannot be reconciled with the claim that there

are exactly as many possible angels as there are alephs.28

The problems facing Lewis and Williamson are highly analogousand bear emphasis. In both cases, we are given philosophical systemsthat allow for quantification over all possible objects. In Williamson’scase this is because all possible objects are actual. In Lewis’ case this isbecause we are allowed to open up our quantifiers so that they rangebeyond what is actual. In both cases, this kind of quantification, whencombined with Indefinite Extensibility, brings proper-class-many pos-

sible angels within the domain of our broadest quantifiers. And thisspells trouble when combined with various of our three ideas.

12. Confining composition

In this and the next two sections, we address possible ways that one orother or our metaphysical targets — Lewis and Williamson, that is —

28 A similar problem arises for someone who wants to combine Indefinite Extensibility with

the assumption that possible worlds are themselves objects — as opposed to properties, pro-positions, or any other items not in the range of the first-order quantifiers. If for each , there

is a world in which there are @

angels, then there are as many possible worlds as there are

alephs, namely, proper-class-many. But this fact cannot be reconciled with the combination of

limitation of size and our mereological result.

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may try to deal with the argument from Limitation of Size. It goes

without saying that those who are thoroughly convinced by the argu-

ment from the Urelement Set Axiom will not, in any case, be assuaged.

When confronted with the Limitation of Size argument, one naturalescape route for Williamson is to restrict the scope of our favourite

mereological principles to concrete existence: while there may well be

a fusion of angels, there need not be a fusion of possible angels. Onenatural restriction — if it even serves to be so called — on mereology is

to first-order objects, objects that are fit assignments for first-ordervariables.29 But that does not help here, since angels are obviously

first-order objects. Similarly, it does not help anyone who takes pos-

sible worlds to be first-order objects. A more relevant consideration infavour of such a restriction is that it may, in any case, be recom-mended by considering the case of the pure sets themselves. Afterall, the combination of limitation of size and our mereological

result cannot be reconciled with our allowing unrestricted composi-

tion to extend even to the realm of pure sets, at least on one natural

assumption on their mereological structure. Assume (with Lewis, forexample) that the plurality of singletons of pure sets is disperse. Then,

by the mereological result, there are strictly more fusions of singletonsof pure sets than there are singletons of pure sets. But this contravenes

the von Neumann limitation of size principle as the singletons of all

pure sets do not themselves form a set and must thereby be in one-onecorrespondence with all things. A natural reaction to this is to delimit

the scope of composition. But once we recognize that composition

does not apply to sets, it may also be natural to remove other non-

concrete objects from its scope of application. Note that this move isnot available to the modal realist. On Lewis’s view possible angels are

just as concrete as actual ones.

29 Note that this provides an attractive way of blocking certain arguments that put prima

facie pressure on limitation of size. Consider for example an argument that proceeds via

noting that there are more identity properties based on the alephs than there are the alephs

(see n. 18). Does this show that the limitation of size principle is false after all? The defender of

limitation of size need not resort to a despairing nominalism at this point. A natural — and we

think more promising — response is, with Frege, to insist on a deep rift between the values of first- and second-order variables, and on the back of this argue that it is mistaken to try to

cram the values of second-order variables into the domain of first-order variables. Assuming a

similar rift between first-order and propositional variables, arguments against the limitation of

size that proceed via the plenitude of propositions can be similarly blocked.





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13. The one and the many

The attempt to make trouble for unrestricted composition by applying

the combination of the limitation of size principle and our mereolo-gical result to the realm of pure sets can be resisted by adopting a non-standard view of their mereological structure. Suppose that one holdsthat a sufficient (though not necessary) condition for an object — setor urelement — to be part of a set is that the former be a member of the latter. Then the disperseness assumption required for the mereo-logical result would not hold. In the case of possible angels, a non-standard merelogy can also escape the argument. When one learns of the Williamsonian framework, it is perhaps natural to think of possi-

bly (but not actually) concrete objects as being scattered like merelo-gical dust across Platonic Heaven. Yet that conception is far fromobligatory. Among the mereological alternatives we suggest one thatis particularly appealing, and which is indeed perhaps more theologi-cally proper. Suppose we adopt a mereology that — as against classicalextensional mereology — abandons the presumption that parthood isantisymmetric: if x and y are parts of each other, x is identical to y .(Such an abandonment is quite familiar among those who are careful

to distinguish a ship from the quanity of steel or wood that constitutesit, even in cases where they eternally coincide). It now becomes pos-sible to think of the possibly (but not actually) concrete objects, not asforming a disperse plurality, but as parts of each other, forming anentangled unity.

When an object becomes concrete it breaks off from — that is,becomes mereologically discrete from those entangled entities, andwhen it ceases to be concrete, it returns to — that is, becomes mer-eologically reconciled to — those entities. While this is not perhaps a

full vindication of Plotinus’ doctrine of a return to the One — carriedinto scholastic philosophy by the early Church Fathers — it is perhapsas close to a vindication as sober analytic metaphysics can provide.

14. Restricting limitation of size

Lewis (1991) opts not quite for von Neumann’s limitation of sizeprinciple, as we have stated it, but instead for a more qualified version

of it, restricted to singletons: a plurality of singletons forms a set, thatis, their fusion has a singleton, if and only if they are not in one-onecorrespondence with all the singletons. So, there is exactly one size aplurality of singletons can be and not form a set. But this leaves

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open whether or not there are more objects altogether than there aresingletons. And indeed, there are, in Lewis’s framework, many morearbitrary fusions of singletons than there are singletons. On this

system, then, the Cantorian sequence of alephs matches the size of the singletons, but it is far from matching the size of all things.

Does this qualification enable him to escape from the Limitationof Size argument that we have advanced against the modal realist?It does not. Lewis is quite explicit that any urelement has a singleton.Assume that there are again exactly as many angels as there are alephs.Then there are strictly more fusions of angels than there are alephs.Since every fusion of angels is itself an urelement, both angels and

their fusions have singletons. But now we are forced to admit —contra the qualified principle — that there are at least two differentsizes a plurality of singletons can have and not form a set. There are, inother words, two different sizes that are larger than any aleph.30

15. A guide for the set-theoretically perplexed

Not all metaphysicians will be content to use pure sets as the basis foran exhaustive account of size. Some will think that pure sets are

merely a useful fiction. Others will indulge in pure set ontology butbelieve that such an ontology is inadequate to a complete account of size. The latter may break from the traditional set-theoretic stricturesby positing a scale of sizes that extends beyond the aleph sequenceby countenancing sizes greater than the size of any set. If the vonNeumann principle holds, this would only add an additional endpointto the series of alephs. But some may postulate, in defiance of the vonNeumann principle, a multitude of sizes beyond the sizes captured

by the alephs.Let us now approach the metaphysics of size in a way that is neutralon set theory. We take a page from Frege, who famously offered aquite a different foundation for arithmetic. He started from what hascome to be known as Hume’s principle :

The number of Fs is identical to the number of Gs if and only if there areexactly as many Fs as there are Gs.

30 To be sure, if we take ‘urelement’ to mean ‘non-set’ — as we have done — then Lewis

would not accept the principle that there is only one size a plurality of urelements can have

and not form a set. (Compare, for example, the plurality of all proper classes, i.e., fusions of

singletons without a singleton, with a subplurality of that plurality of the size of the pure sets.)





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One immediate consequence of Hume’s principle is that no matter

what Fs we consider, we are in a position to ascribe a cardinal number

to them — whether or not they are set-sized. Thus even if there is no

set of all self-identical objects, we can, according to Frege, assigna cardinal number to them, that is, anti-zero.31

In what follows, we will reify sizes as certain objects, but, unlike

the traditional set-theoretic framewok, we will assume them to begoverned by a modal version of Hume’s principle as formulated in

a suitable language:

Necessarily, the size of Fs is identical to the size of Gs if and only if thereare exactly as many Fs as there are Gs.

Let us concede upfront that the adoption of this principle relies on thepresupposition that any two pluralities are comparable in size: either

there are exactly as many Fs as there are Gs, that is, they are in one-one

correspondence, or there are strictly fewer Fs than there are Gs, that is,the Fs are in one-one correspondence with some (but not all) Gs, or

there are strictly more Fs than there are Gs.32 This assumption takes us

far beyond orthodox set theory, which remains silent when it comes to

the question of whether various proper classes are comparable in size.

Some further assumptions are natural enough, though not extrac-table from the principle alone. Some of them are analogues to certain

unofficial assumptions on the nature of sets. In particular, it is natural

to assume that sizes, even if not sets, are, like sets, necessary beings.We assume, too, that it is not necessary that everything is a size.

Cardinal comparability gives us that the sizes posited by our prin-

ciple are linearly ordered. A further natural assumption — though

certainly one not extractable from the previous ones — is that sizes

are well ordered, i.e., given any sizes, there is a least one of them. (Thisrules out, for example, the hypothesis that not only is the Continuum

Hypothesis false, but also there is a countable number of densely

ordered sizes between the size of the reals and the size of the naturals.)In a setting like this, where an ontology of sizes is posited that need

not be identified with sets, there is little to speak in favour of the

Urelement Set Axiom. If there are no sets, it will not be true. And

even if sets are posited alongside sizes, the Urelement Set Axiom

cannot be tolerated since it will be manifestly unacceptable to allow

31 The term ‘anti-zero’ comes from Boolos (1997).

32 Otherwise, if too many pluralities turned out not to be comparable in size, then there

might not be enough objects to satisfy the principle in the first place.

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that the sizes form a set. (Otherwise, the set of all sizes would have an

aleph as its cardinal number. But this means that our scale would leaveout sizes for any succeeding aleph, let alone the entire series of

alephs.)33 And in a setting like this, there is little force to the vonNeumann principle. When the aleph series no longer forms thebasis for the metaphysics of size one cannot, without special additionalargument, plausibly assume that the alephs arbitrate the limitations

on size.What are the alternatives to Indefinite Extensibility and Plenitude in

this new setting? And how should we choose among them?We have been silent on one feature of the linear ordering of sizes,

namely, whether it has an absolute upper bound, a modal anti-zero.To be sure, our modal variation of Hume’s principle requires the

existence of an anti-zero relative to each world in which it holds.But if the universe is modally inconstant, that observation does notsettle whether there is a modal anti-zero. We may, after all, wish the

scale of sizes to mark all the alternative sizes some objects might have,whether or not they have do them. But then, what counts as anti-zerowith respect to the actual world may not correspond to the endpoint

of the entire scale of possible sizes.Now if sizes are themselves objects and, as we have assumed, them-

selves necessary beings, then the size of the objects in each world is atleast the size of all the sizes. Given that there is at least one non-size,there are infinitely many sizes. But little further can be deduced from

the preceding assumptions, and, in particular, we cannot deducewhether or not the well-ordering of sizes has an upper bound.

Let us now return to our original question both from within a

framework that assumes no end-point for the series of sizes and one

that does.A structure with no endpoint is notably different from the com-

bination of the Cantorian framework and the von Neumann principle.There is manifestly a topmost size — at least if we assume modalconstancy for the alephs and hence the pure sets. But in the present

setting there is no such size. If sizes have no endpoint, no naturalanalogue for Plenitiude is available. Of course one could still insistthat, even though there are plenty of sizes greater than any aleph,

the maximal size for angels is that of the entire aleph sequence.But what could justify this when there are plenty of sizes beyond

33 We understand ‘urelement’ to mean ‘non-set’ and leave open whether the urelement set

axiom could be tolerated on a different reading of the term.





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that of the aleph sequence? Here, then, the natural view is an analogue

of Indefinite Extensibility: no matter what size you pick, there couldhave been strictly more angels than that on the point of a needle.

But notice now that there are crushing objections to Lewis andWilliamson’s modal metaphysics, and indeed, to any metaphysicsthat quantifies over all possibilia. For if, for any size, there could bethat many angels at the end of a needle, then no size whatever can becoherently associated with the merely possible angels (in Williamson’scase) or with the transworld panoply of angels (in Lewis’s case). Thelesson is that if one is to indulge in quantification over all possibiliathen one had better opt for a scale of sizes with a topmost element.

Suppose instead one opts for a scale of possible sizes with a topmostelement. Let us call the topmost size, a modal anti-zero. The necessi-tarian is independently committed to there being such a size, which ismodally constant. Then one can adapt two of the earlier positions tothe present setting. One position is that the number of angels on thepoint of a needle could be anti-zero — the analogue of Plenitude.The other is that while the number of angels on the point of aneedle could not be the modal anti-zero, there is no upper bound

on the size angels could be — and, indeed, for any size less than anti-zero, the angels could be strictly larger in size than . This is theanalogue of Indefinite Extensibility. But given that there are strictly more fusions of angels than there are angels, the analogue of Plenitudewill be unstable. And, as before, the analogue of Extensibility will berendered unstable by various modal metaphysics that permit quanti-fication over all possibilia.

However, in the current framework there is a genuinely new option.In the Cantorian framework combined with the von Neumann limita-

tion of size principle, the maximal cardinality — namely, that of theentire aleph sequence — has no immediate predecessor. However, we

are not, in the current framework, entitled to the assumption that the

modal anti-zero has no immediate predecessor. Of course we may

boldly speculate that the modal anti-zero does not have one. But it

is at least open to us to speculate that the modal anti-zero does have

an immediate predecessor, call it a modal anti-one. In this setting we

may have a principled reason for claiming that there is a particular

maximum less than the modal anti-zero that angels can have, namely,the modal anti-one. Our mereological result will rule out the modal

anti-zero, but not the modal anti-one. (Note that the mereological

result cannot be reapplied to the fusions of the angels to get an even

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larger size since the discreteness assumption will not hold for the

fusions.)34

16. A meta-ontological sermon

Some lost souls, rebelling against the natural light, have promulgatedthe heretical view that metaphysics is but wordplay. What went by thename of substantive metaphysics among our distinguished semanticpredecessor was, according to these naysayers, a case of capturing the

same facts using different notations. What was taken to be substantiveis alleged to be superficial. And what goes for our predecessors goes forcontemporary metaphysicians who carry the torch of metaphysicallearning. These lost souls are careful not to avow the heresies of verificationism, at least not in public. Instead, they avow what pur-ports to be a new semantic picture, one according to which there isno distinguished or privileged meaning of ‘exists’, ‘there is’, ‘someobject’, or of any of the other foundational expressions in whichontological disputes are canvassed. Instead, ontological proposals, if they are not to be construed as crazy disavowals of what is self-

evidently true in English, are at best charitably construed as tacitproposals for using ontological language in new ways to capture oldfacts. We cannot engage directly with the detailed semantic theories

that these naysayers offer us. For they have not provided such theories.Instead, they present us with a ramshackle mixture of Mooreanposturing and sample translation schemes that allegedly render meta-physical questions empty.

The paradigm example — indeed, often the only example that is

offered — of an empty metaphysical dispute is that between the nihi-list who believes that only simples exist and the proponent of classicalmereology who adds a full stock of fusions to whatever simples exist.At least part of the rhetoric of our heretical opponents is that every possibility — construed as a set of possible worlds — that is embracedby the proponent of one position is embraced by the other. Let us say that position A modally advances on position B iff there is some set of worlds such that there is a sentence in A’s language that is true at alland only those worlds and which is reckoned to be possibly true by the

34 If with Lewis we admit singletons, there may be a principled reason for a third answer, a

modal anti-two. Moreover, if the power-plurality operation jumps n steps up the size tree, we

may envisaged reason for other answers as well namely, a modal anti-three, a modal anti-four,

etc.





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defender of A, but there is no sentence in B’s language true at alland only those worlds and that is reckoned to be possibly true by the defender of B. With this in place, one might think it altogether

obvious that, assuming the Nihilists language is sufficiently rich, andinterpretation is sufficiently charitable, the Mereologist and Nihilistwill be so situated that neither makes a modal advance upon the other.Insofar as one agrees that there is a non-empty set of worlds pickedout by ‘there are three things’ in the mouth of the Mereologist, itwill be a set captured by ‘there are two simples’ in the mouth of the Nihilist (and will also be captured by ‘there are two things’ inthe mouth of the Nihilist on the charitable assumption that the

Mereologist ought to treat him as using a quantifier that is restrictedrelative to the Mereologist’s quantifier).

Now there is plenty to say about this sort of semantic picturethinking which goes beyond the scope of this tract. But one lesson

should be immediately obviously from the considerations adduced

above, namely that it is far from clear that mereology makes no

difference to which possibilities are possible for the concrete simples

themselves. In particular, suppose that the mereologist adopts a con-

ception of size grounded in the Cantorian framework. He will likely,

for the reasons given, balk at the possibility of a plurality of concrete

simples that are as plenitudinous as the alephs. By contrast, the

Nihilist will be under no similar pressure to deny that possibility.

Hence, from the perspective of the Nihilist there will be a genuine

possibility for the concrete simples that the Mereologist is blind to —

on any reasonably charitable and natural construal of the

Mereologist’s language. Thus, the Nihilist will take himself to have

modally advanced on the Mereologist. Of course things will look

different to the Mereologist. If he charitably construes the Nihilist’squantifiers as restricted, he will not think that the Nihilist is guilty of

error in the claim there are only simples. But he will, even from this

(excessively?) charitable perspective construe the Nihilist as incorrect

in claiming ‘Possibly, the concrete simples are so numerous as to

match the alephs in size’. From the perspective of the Mereologist,

there is no modal advance made by the Nihilist. So, does one theory

modally advance on the other? One can only decide this by figuring

out which theory is true! When the deconstructionist smoke hascleared, we can only settle whether one theory modally advances

on the other by theorizing as best we can. The lesson is obvious

enough: semantic ascent to a discussion of language games, notations

How Many Angels? 79




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and linguistic rules was never an adequate substitute for doing

metaphysics.35

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35 We would like to thank Øystein Linnebo, Daniel Nolan, Robbie Williams, Timothy

Williamson, and audiences at Abeerden, Chicago, Leeds, Manchester, MIT, Nottingham,

Oxford, Paris, and Vancouver.





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