Conceptions of Set

Luca Incurvati’s Conceptions of Set, 7

In a couple of very well known papers, George Boolos argued that “the axioms of replacement do not follow from the iterative conception”. Was he right? Or can Replacement be justified on (some core version of) the iterative conception? This is the topic of the particularly interesting §3.6 (pp. 90–100) of Conceptions of Set, ‘The Status of Replacement’.

Luca discusses three lines of argument to be found in the literature for the thought that the iterative conception does warrant Replacement. I’ll comment on two in this post.

The first he calls Gödel’s Argument. Two quotes from Gödel: (i) “From the very idea of the iterative concept of set it follows that if an ordinal number α has been obtained, the operation of power set P iterated α times leads to a set Pα(∅)” And then (ii) “the next step will be to require that any operation producing sets out of sets can be iterated up to any ordinal number.”

In response, Luca makes the following central points:

  1. Tait and Koellner have argued that elaborating Gödel’s claim (i) requires appeal to Choice. But not so. For we can work with the Scott-Tarski definition of an ordinal, and then, without needing an assumption of Choice, Gödel’s thought will at least warrant adding to Z+ the Axiom of Ordinals — the axiom that there is a level Vα for every ordinal α.
  2. This theory with the Axiom of Ordinals is rich, much more powerful than Z+, and in fact buys us the nice results that Boolos claimed for Replacement. However, the Axiom of Ordinals is weaker than full Replacement.
  3. A version of Gödel’s second claim (ii) is needed to get us from the iterative conception to full Replacement, and it isn’t clear why (ii) should be thought of as part of the iterative conception.

On (1), accepting Gödel’s (i), Luca’s discussion seems spot on. On (2) quite a few readers (those familiar with ZFC but who haven’t read Potter’s book) might well have welcomed rather more at this point on the Axiom of Ordinals, on its virtues and mathematical consequences. They might reasonably ask: if the Axiom of Ordinals in fact gives us (a good deal of) what we want, just why — other than conservative adherence to tradition! — should we buy full-power Replacement? On (3), we could I suppose go back to pause over (i) to wonder if the idea of the universe of sets being layered by iterating the set of operation has to go along with the idea that those iterations should be ordinal-many (for any ordinal we can obtain). But leave that aside: Luca it is surely right that it is one thing to build into the iterative conception the idea that the core set of operation should in some sense be iterated ‘as far as possible’, it is another thing to require that other operations be iterated as far as possible too.

The second line of argument for Replacement to be discussed is what Luca calls The Argument from Co-finality. Thinking in terms of stages of the hierarchy, Shoenfield (in his famous Handbook article) suggests that for any collection of stages S, there will be one after it “provided that we can imagine a situation in which all of the stages in S have been completed”. But then assume that we have a set x and for every y in x there is a stage Sy correlated somehow or other with y. Then

Suppose … we take each y in x … and complete the stage Sy. When we reach the stage at which x is formed, we will have formed each y in x and hence completed each stage Sy.

So if S is the collection of stages Sy, we can imagine a situation in which all of the stages in S have been completed and there will be a stage after it.

Now, if we buy this cofinality principle, then Replacement is immediate (since Replacement tells us that the image of x under a function f will itself be a set; for each y in x take Sy to be the stage at which f(y) is formed, and then by Shoenfield’s principle there will be a stage after all those at which we can gather together all the f(y) into a set …).

What are we to make of all this? Luca surely hits the nail on the head! As he neatly notes, Shoenfield’s argument

seems to assume that if a process can be completed, and we replace each stage of the process with a process that can be completed, then the maxi-process consisting of all these processes can itself be completed. But this is just the Axiom of Replacement reformulated in terms of stages and processes.

The supposed defence of Replacement is therefore too close to being circular.

So we haven’t got here an independent argument from the iterative conception to Replacement. Luca concludes, however, on a (surprisingly?) sympathetic note: “the cofinality principle certainly seems to harmonize well with the iterative conception, and can perhaps be seen as one way of spelling out the idea that the cumulative process through which the hierarchy is obtained should be iterated as far as possible.” But equally couldn’t we spin it the other way? — the iterative conception itself doesn’t take us as far as Replacement, and it takes a further independent thought to justify that principle. Which leaves us with the question of what rival further thoughts (equally harmonizing with the basic iterative conception) might be on the cards.

To be continued on a third line of argument for Replacement, via reflection principles.

Luca Incurvati’s Conceptions of Set, 6

In §3.3 of Conceptions of Set, Luca discusses what he calls the ‘no semantics’ objection to the iterative conception. He sums up the supposed objection like this:

Consider the case of iterative set theory, which for present purposes will be our base theory Z+. Since the set-theoretic quantifier is standardly taken as ranging over all sets, it seems that one of the interpretations quantified over in the definition of logical validity for L [the standard first-order language of set theory] – the intended interpretation – will have the set of all sets as its domain. But there can be no set of all sets in Z+, on pain of contradiction. Hence, the objection goes, if we take all sets to be those in the hierarchy, we cannot give the usual model-theoretic definition of logical validity.

Or rather, that is how the objection starts. Of course, the further thought that is supposed to give the consideration bite is that, if we can’t apply the usual model-theoretic definition of logical validity, then we are bereft of a story to tell about why we can rely on the inferences we make in our set theory when we quantify over all sets.

As Luca immediately remarks, this challenge is not especially aimed at the iterative conception: any conception of the universe of sets that rules out there being a set of all sets will be open to the same prima facie objection. It looks too good to be true!

Graham Priest is mentioned as a recent proponent of this objection. But as Luca point out, Kreisel over fifty years ago both mentions the issue raised in the quote and has a response to what I called the further thought which is supposed to make the issue a problem. For Kreisel’s ‘squeezing argument’ is designed precisely to show that we have a perfectly good warrant for using standard first-order logic as truth-preserving over all structures, not just the ones that can be formally regimented in the usual model-theoretic way. I’ve defended Kreisel’s argument, properly interpreted, e.g. here: so I’m more than happy to go along with Luca’s endorsement.

Luca does, however, have other things to say about the ‘no semantics’ objection before turning to Kreisel’s way out. As he notes, we can hold onto the idea that we can sensibly quantify over all sets, and hold on to the core of the classical Tarskian definition of validity, by denying that domains have to be taken to be sets. Of course, there is little point in arm-wavingly talking about classes instead of sets as if that by itself resolves anything. What we need to do is to reject what Richard Cartwright calls the All-in-One Principle which tells us that to quantify over some things presupposes that there’s a set (or proper class, or other single object-in-its-own-right) to which they all belong. We can speak, if you like, of virtual classes, classes-as-many, i.e. we can use a singular idiom for talking about objects, plural. But better, we should just go straight to giving the semantics for FOL in a plural metalanguage, saying that quantifiers range over objects (one or many), interpretations assign e.g. monadic predicates some of these objects (zero, one or many), and so on. We know this can be done, and the plural metalanguage itself formalized — Oliver and Smiley show how do this in all the detail you could want in their Plural Logic. I’m really not sure why Luca doesn’t mention the possibility of taking the plural route [added meaning something like the Oliver/Smiley version] here, as it would surely give him [added a simple and direct] rebuttal of the further thought that drives the ‘no semantics’ objection.

What Luca does discuss is another way of giving up the idea that the domain of quantification is an object which he finds in work by Rayo, Williamson and Uzquiano. They propose a higher-order semantics where our metatheory is to be regimented in a second-order way. [Added The second-order semantics, as Luca points out, can be given a  plural interpretation, but I don’t find the second-order version here] especially natural, and it is not at all clear to me why going via higher-order logic should be thought a better bet than staying first-order but allowing plurals.

Luca Incurvati’s Conceptions of Set, 5

We have now reached Chapter 3, which can be conveniently divided into three parts. The first part (§§3.1–3.2) discusses two initial, and very general, challenges to the iterative conception, challenges which (it seems to me) can be fairly readily met. These are the topic of this post. The second part of the chapter (§§3.3–3.5) discusses another very general challenge, to my mind a rather more interesting one: I’ll consider that in my next post. The final part (§3.6) discusses a more specific challenge (meaning one that arises from focused technical questions, about the status of replacement, rather than from sweeping conceptual considerations). I’ll need to revise my homework on replacement! — but hope to have something sensible to say in a third post. So to begin …

The first challenge to the iterative conception we’ll discuss is what Luca calls the missing explanation objection. In brief,

[I]f we take all sets to be those in the hierarchy, we cannot explain the appeal of the naïve conception of set, as embodied in the Axioms of Comprehension and Extensionality.

This supposed objection has been pushed by Graham Priest, but it has always struck me as pretty feeble. Assume we have distinguished the concept of set (a unique object over and above its members) from the idea of a class-as-many. Now we have this intended concept of set in play, there is room to further distinguish the following two claims: (i) [Naive property comprehension] for any property, there is a set of all and only the objects with that property, and (ii) for any determinate plurality of objects, there is a set of all and only them. Now, (i) gives us e.g. a set of all sets, while (ii) doesn’t — because no determinate plurality of sets is all the sets (since given that plurality of sets there is, by (ii), another set, namely the set of them). The defender of the iterative conception, who will reject (i) but can accept a version of (ii), can then reasonably say that once we’ve distinguished sets from classes-as-many, the remaining appeal of (i), such as it is, comes from confusing it with (ii). And once the distinction is made and properly grasped, the appeal should vanish.   [Of course, this isn’t to say that the iterative conception is definitely right, or that the idea of a universal set is definitely wrong: the challenge though was that the iterative theorist had nothing to say about the appeal of the naive conception — and that seems wrong.]

This, at any rate, is the sort of story I would have given. Luca says rather more over six and a half pages. In the first past of his discussion, he presses the distinction between (i) and the claim (i*) for any property of individuals, there is a set of all and only the individuals with that property [where the individuals are the non-sets]; and he suggests that part of the appeal of (i) comes from confusing it with, or recklessly generalizing from, the harmless (i*). Perhaps there is something in that, though I’m not very sure. The second part of Luca’s discussion then gives a more careful treatment related to — though not, I think, quite the same as — the response that I sketched.

The second challenge to the iterative conception, again pressed by Priest but also encountered elsewhere is what Luca calls the circularity objection. This arises from the suggestion that iterative conception of the set-theoretic is “parasitic on a prior notion of an ordinal” and, if we are not going to go round in circles, that’s will need to be derived from a different notion of set (so the iterative conception can’t be fundamental).

But this too has always struck me as feeble (roughly: it depends on forgetting that the von Neumann ordinals are a handy implementation, not the one-and-only possible story about ordinals-as-indexers-for-tranfinite-processes). After all, we can get a long way into the theory of at least countable ordinals without talking about sets at all — we just need numbers (as individuals) and order-relations on them. If you insist on treating relations as sets of pairs which are themselves sets of sets, you still only need a few levels of sets. So: start with the numbers and a few levels. Develop a theory of countable ordinals. Use them to index more levels (lots of levels!) to get a very rich universe. In this universe we can define many more ordinals. OK, so we can now lever ourselves us by indexing more levels with these new ordinals. And so on upwards … There’s no circularity. When adding stages of the hierarchy, we already can define the ordinals we need to index the additional stages. This sort of idea was already being explained by Gödel in 1933.

Ok, that’s a bit arm-waving, but they basic idea is probably familiar. Turning to Luca’s discussion, he first gives a considerably more careful and more developed version of this Gödelian pre-emptive response to the challenge.

But he then adds a very important additional point which is worth highlighting here:

The axiomatization given by Scott (1974), of which [Scott-Potter] SP is a descendant, shows that the worry that the notion of a well-ordering is needed to grasp the iterative conception is really just an idle concern. In particular, what Scott provided is an axiomatization of set theory which, albeit sanctioned by the iterative conception, does not assume a previous conception of the hierarchy as constituted by levels ordered by the ordinals. Rather, starting from certain elementary facts about levels, which … he called ‘partial universes’, he established facts about sets and levels. Notably, what is assumed about the levels does not include that the levels are well-ordered. More specifically, he showed that, assuming the Axioms of Restriction and Accumulation, we can prove, together with the Axioms of Separation and Extensionality, that all axioms of Z except for Infinity hold, that every set is well-founded and, crucially, that the levels are well-ordered by membership. … The upshot is that the fact that the levels of the hierarchy are well-ordered is not required to grasp the iterative conception, but is a consequence of it. I conclude that we do not need a prior and different notion of set to make sense of the notion of the cumulative hierarchy, and the circularity objection fails.

That seems conclusive.

Luca Incurvati’s Conceptions of Set, 4

On now to the second half of the second chapter, where we are still considering the iterative conception in an initial way. 

So, quoting Luca,

According to the iterative conception, then, sets can be arranged in a cumulative hierarchy divided into levels. This conception sanctions (at least) most of the axioms of standard set theory and provides a convincing explanation of the paradoxes; but is it correct?

What reasons, then, can be offered in support of endorsing the iterative conception?

Luca first discusses the idea that we should take literally the metaphor of construction that comes to us so readily in describing the iterative conception. So, the idea is, sets really are formed in a stage-by-stage process, where at each stage we can only collect together in various ways what is already available.

But how do we make better-than-metaphorical sense of this idea of forming sets in a process when we are supposed to be dealing (aren’t we?) with abstract items which (i) exist independently of our activities (aren’t really formed) and (ii) in a timeless way (so there’s no real process of level-building). Arguably, the constructionist metaphor at best gives colour but no real underpinning to the iterative conception.

Suppose, however, we do try to push the metaphor harder. Then, Luca argues, [my numbering]

(i) it seems part of the constructivist doctrine that, at any point in the construction process, we can only construct sets specifiable by reference to sets already constructed. (ii) This, however, seems to sanction only a predicative version of Z’s Separation Schema …

which cuts down the strength of our set theory. Now, (i) gives us one way of elaborating what the ‘the constructionist doctrine’ might be supposed to be. Though we could, I suppose, pause to ask whether is it compulsory to construe ‘construct sets from sets that are already constructed’ as implying ‘construct sets specifiable by reference to sets already constructed’. Be that as it may, it would have been good if Luca had then paused longer over the implications of (ii), saying more about predicative set theories. Just how weak are they? If we can live with weak predicative set theories for ordinary mathematical purposes (as Feferman claimed, for example), then why not treat them as what is, on second thoughts, warranted by a rather strictly interpreted iterative conception? Some readers might have wanted rather more here.

However, with the iterative conception so understood, we’ll have to back off from our original thought that the iterative conception sanctions (most of) standard set theory. And Luca takes this in itself to be a reason to resist the constructivist gloss on the iterative conception.

Moving on — we’ve got to §2.4 of the book — Luca next considers the idea that we can underwrite the iterative conception, not by saying that the sets are literally ‘formed’ stage by stage, but by invoking a [now timeless] relation of metaphysical dependence between a set and its members: the hierarchy reflects this structure of metaphysical dependence.

Not surprisingly — or at least, not surprisingly to someone as sceptical about such metaphysical notions as I am — Luca has little trouble in showing that various attempts to elucidate this supposed relation of metaphysical dependence in terms of other metaphysical notions (like that of essential property) either go round in very tight circles, or pretend to explain the obscure in terms of the even more obscure. Moreover it is quite unclear, as Luca also argues, that even if we could make good a suitable notion of metaphysical dependence, that this would underpin an iterative hierarchy of the right structure (can’t there be, for a start, circles of metaphysical dependencies?). The critical discussion in §2.4 seems pretty conclusive to me.

So where does that leave us? We can’t, it seems, underwrite the iterative conception (or at least an iterative conception that will sanction something like standard set theory) by trying to cash-out a construction metaphor or a metaphor about dependence. But then recall this well-known remark from George Boolos about the iterative conception, aptly quoted by Luca:

[F]or the purpose of explaining the conception, the metaphor is thoroughly unnecessary, for we can say instead: there are the null set and the set containing just the null set, sets of all those, sets of all those, sets of all Those, … There are also sets of all THOSE. Let us now refer to these sets as ‘those’. Then there are sets of those, sets of those, … Notice that the dots ‘…’ of ellipsis, like ‘etc.,’ are a demonstrative; both mean: and so forth, i.e. in this manner forth.

Luca picks up on Boolos’s thought, and argues that we should indeed be content with what he calls a minimalist account of the iterative conception (we are supposed to hear echoes here of talk about a minimalist account of truth — I’m not entirely convinced that’s helpful, given that minimalism about truth is deflationist in spirit while Luca’s iterative conception remains very robust; but let that pass). He finds such a conception already in Gödel, quoting a remark where he talks of a concept of set

according to which a set is anything obtainable from the integers (or some other well-defined objects) by iterated application of the operation (“set of”).

And that, the suggestion goes, is where the iterative conception bottoms out, just in the idea of iterating applications of ‘set of’ (where the result of an application is distinct from any of the things it is applied to).

Note, in passing, that if what crucially matters is the set of operation, and (as Boolos’s words indicate) this operation takes zero, one, or more things (plural), and yields a set of them, then arguably the natural logical home for set theory would seem not to be standard first-order logic which has no place for plurals (no formal equivalent of ‘those’!); rather it seems we will want a plural logic which can treat operations mapping many to one. We’ll have to see if this thought is taken up later.

Anyway, Luca proposes that we take the iterative conception ‘neat’ (so to speak), without the supposed support of further thoughts about constructions or dependencies. But without those illusory further supports, why should we think it is a good conception? Well, this is what the rest of the book is going to be about … showing on the hand that other conceptions won’t give us what we want, and on the other hand that the iterative conception, minimally construed, can resist various critical attacks. So we get (in Luca’s words) an ‘inference to the best conception’. We’ll have to see how this promised inference pans out!

To be continued: the next chapter is on Challenges to the Iterative Conception.

Luca Incurvati’s Conceptions of Set, 3

Well, I’m half-way through the task of writing up answers to the Exercises for Chapter 41 of ILF2 (and since I have the space for a few additional exercises, I’ll be trying to think up some more). But there is only so much excitement I can take! So let me return for a bit to reading Conceptions of Set. And by the way, do note that Luca has now commented on my first tranche of comments.

Chapter 2 is called ‘The Iterative Conception’, and really divides into two parts. The first part outlines this conception (and explains its relation to [some of] the axioms of set theory). The second critically considers whether the conception can be grounded (as some have supposed) in the thought that there is a fundamental relation of metaphysical dependence between collections and their members. More on this very interesting second part in my next posting. For now, let’s just think a bit about the iterative conception itself, mention some issues about the height and width of the cumulative hierarchy, and then say something about some set theories which tally with this conception.

Luca’s discussion starts like this:

On the iterative conception, sets are formed in stages. In the beginning we have some previously given objects, the individuals. At any finite stage, we form all possible collections of individuals and sets formed at earlier stages, and collect up the sets formed so far. After the finite stages, there is a stage, stage ω. The sets formed at stage ω are all possible collections of items formed at stages earlier than ω – that is, the items formed at stages 0, 1, 2, 3, etc. After stage ω, there are stages ω + 1, ω + 2, ω + 3, etc., each of which is obtained by forming all possible collections of items formed at the preceding stage and collecting up what came before. …

Of course, that’s exactly the usual story! But perhaps we should discern two thoughts here. There’s the core iterative idea that sets are built up in stages, and that after each stage there is another one where we can form new sets from individuals and/or the sets we have formed before. This captures an idea of indefinite extensibility, while rejecting the idea that at any stage we have formed all the sets (so we develop this thought, it looks as if we are going to avoid entangling ourselves with the familiar paradoxes). Then we have the further idea that we can iterate the set-building transfinitely; there are set-building stages indexed by limit ordinals, where we can collect together everything formed so far.

Luca of course stresses that the iterative conception itself leaves it open how far the cumulative hierarchy goes (what the ‘height’ of the universe is). But I think he is more concerned with how far into the transfinite we should go, while I would have liked him to pause longer here at the start, over the question of why we need to go into the transfinite at all. After all, it might be said, if we are allowing individuals, then a set universe where we have the natural numbers as individuals and then the finite levels of the hierarchy gives us a capacious setting in which arguably most mathematics can be carried out. So someone might ask: why commit ourselves to more, why go transfinite? But we’ll no doubt be coming back to issues of ‘height’

The iterative conception also leaves it open what exactly we are to make of forming ‘all possible collections of items’ from earlier stages. How ‘wide’ or ‘fat’ is each stage? ‘All possible’ certainly seems intended to be more generous than e.g. ‘all describable’; which is why we think the axiom of constructibility V = L gives us a cumulative hierarchy of rather anorexic stages, less than we intended, and why the axiom of choice can seem so natural. We are tempted to say: if all (banging the table, yes ALL!) sets are formed at each stage, then surely the needed choice sets are formed in particular. But as Luca nicely points out, following Boolos, that tempting thought is on second thoughts not so convincing, unless we build in another thought which is not itself part of the core iterative conception. The extra we seem to need is the combinatorial conception’s thought that “the existence of a set does not depend on the existence of a condition satisfied by all the members or of a rule for selecting them, [so] nothing seems to stand in the way of the choice sets being formed”. But again, we’ll need to come back to issues of ‘width’.

And what about the individuals at the ground level of the hierarchy? Do we need to consider set theories with urelements? Luca makes a familiar point:

From the mathematician’s perspective, starting with no individuals makes a lot of sense: mathematicians tend to be interested in structures up to isomorphism, and it is usually assumed that — no matter how complex or big a putative set of individuals might be — there will always be a corresponding set in the hierarchy of the same [size].

(Actually, Luca writes ‘order type’ rather than ‘size’; I’m not sure why.) So for many mathematical purposes we can do without individuals, and Luca proposes to typically focus his attention on pure set theories.

OK, so far so good: now turn to the question of what set theories the iterative conception might give its blessing to.

There are familiar worries about replacement and choice, so Luca shelves those for later consideration. And set aside extensionality as already underwritten by our very concept of set. Then Luca argues — in a familiar way — that the iterative conception sanctions the other axioms of Zermelo set theory Z. But he discusses other theories too: the stage theory ST of Shoenfield and Boolos; the theory Z+ which you get by replacing the Axiom of Foundation with an axiom which asserts that every set is the subset of some level of the hierarchy; and SP (a version of) Scott-Potter set theory. Luca argues, plausibly enough, that the iterative conception underwrites not only ST (which implies the axioms of Z leaving aside extensionality), but also Z+ and SP (those two theories in fact being equivalent).

Those claims are all persuasive. If I have a comment, then, it is about presentation rather than content. Luca’s Chapter One finishes with a couple of Appendices, two pages on cardinals and ordinals, Cantor/Frege/Russell vs the standard ZFC treatment, and one page on Cantor’s Theorem. Fine. But if a reader needs those explanations of some absolute basics, then I suspect they are going to need significantly more explanation here. For many a reader will only have encountered standard Zermelo Fraenkel set theory, and would surely have welcomed a less rushed treatment (or another chapter Appendix) elaborating on those neighbouring alternatives — especially given that some of these embody the iterative conception in a particularly direct and appealing way.

To be continued, with a discussion of Luca on grounding (or not grounding) the iterative conception in some idea of collections ‘depending’ on their members.

Luca Incurvati’s Conceptions of Set, 2

We are still on Chapter 1 of Luca’s book. Sorry about taking longer than I had intended to get back to this. But I’d promised myself to get the answers to the Exercises for Chs 32 and 33 of IFL2 (on natural deduction for quantifier arguments) done and dusted. Thirty eight pages(!) of work later, they are online!

Let’s take it that the concept of set is (at least in part) characterized by Luca’s three conditions — Unity (a set is in some sense a unity, distinct from its members), Unique Decomposition (a set decomposes into its members in just one way), Extensionality.

Which leaves more to be said, no doubt. But then there are various possible views of the role of the further story we need.

Suppose, for example, that you hold that the concept of set, as pre-theoretically grasped, is governed by the following assumption: that for any coherent predicate there is a set of objects which satisfy it. Then, rapidly, we get to a classically inconsistent naive set theory. Put on hold for now the option of revising your logic as a palliative. Then you’ll want to work with a classically consistent replacement concept of set*. And the further story we need is an elaboration of this replacement concept.

Suppose alternatively that, as far as it goes, the concept of set is consistent enough. Then that leaves open a spectrum of possible views (at least I take it there is a spectrum here, though Luca highlights the endpoints). At one end, the idea will be that there is not much more to be said about the basic concept of set. We can go on, though, to sharpen the notion in a number of distinct ways, coming up with different, more refined, concepts — though it may turn out that one sharpening is particularly fruitful, mathematically speaking. [Possible model: we have a rough-and-ready concept of a computable function. This can be refined in various ways, though one direction — giving us the notion of an effectively computable function, where we abstract from considerations of computation length or storage costs, etc. — turns out to be particularly fruitful.]

At the other end of the spectrum, the idea will be that our pre-theoretical dealings with the notion of set reveal our perhaps partial grasp of a single, sharply definite, concept. So now what we need is not a sharpening/refinement/filling-in-of-the-conceptual gaps so much as  an analysis of this concept, a concept which we perhaps initially ‘perceive’ only through a glass darkly. [Here perhaps enchoing Gödel who had some such view of ‘perceiving’ mathematical concepts.]

Now, whether we want something on the sharpening/analysis spectrum or want replacement, Luca talks of this being provided (or at least a start being made) by elaborating a conception of sets — which he characterizes as a (possibly partial) answer to the question what is it to be a set, an answer which “someone could agree or disagree with … without being reasonably deemed not to possess the concept” set.

I’m happy with the spirit of all this, and with Luca’s view that to make progress on the interesting questions, we don’t really need to worry too much whether we are sharpening, analysing or replacing! But I suppose we could niggle about the letter of his discussion. A self-conscious sharpener (we might reasonably argue) isn’t saying what it is to be a set, tout court, but what it is e.g. to be a set in the iterative hierarchy (compare, a sharpener talking about computable functions isn’t saying what it is to be computable, in the one true sense, but e.g. what it is to be effectively computable). Likewise, a replacer isn’t saying what it is to be a set — nothing falls under that inconsistent concept, says he — but rather is saying what it is to be a set*, where this a concept which will actually do much of the work we want in a coherent way. Maybe Luca’s framework gets a bit procrustean here.

But as I say, I’m happy to grant the basic point: there’s a difference between outlining what anyone who counts as having the pre-theoretical concept of set needs to grasp, and going on to articulate a conception of sets in the sense of some guiding thoughts about what sets might be that can shape fully-fledged theory-construction.

In this initial chapter, Luca has something to say about three such guiding thoughts. One we have already touched on, the thought that every contentful predicate has a set as extension, which lands us with naive set theory. Luca then gives a familiar diagnosis of what goes wrong. Say a concept is (i) indefinitely extensible if, taking any set of things which fall under C, there is an operation which produces a further thing which falls under C. Say a concept is (ii) collectivizing [Potter] or universal [Luca] if there is a set of everything that falls under the concept C. Then naive theory takes the concept of set to be both (i) indefinitely extensible and (ii) universal. And that way contradiction lies. A similar diagnosis can be given, as Luca nicely explains, for Cantor’s Paradox, the Burali-Forti Paradox and Mirimanoff’s Paradox. So we want our developed consistent set theory to allow only one of extensibility and universality. And Luca promises to discuss theories of both kinds. And now we see that one role for a conception of sets in the sense of some guiding thoughts can be (i) to indeed guide us in one direction or the other, and (ii) also give us some initial confidence that we are going to avoid falling into paradox.

Luca at the end of his chapter discusses two more guiding thoughts, what he calls the logical and combinatorial conception of sets. A logical conception treats sets as essentially associated with some predicate, concept, or property (the naive theory involves a naive version of this thought). A combinatorial conception arguably goes back to Cantor himself — and can be associated with images such as a sequence of random choices of what goes into the set. Thus Bernays writes that on this conception, “one views a set of integers as the result of infinitely many independent acts deciding for each number whether it should be included or excluded.” Which might well raise some philosophical eyebrows. Indeed there’s a long tradition that worries that standard set theory is [as it were! — see comments] conceived in sin, an unholy alliance between a logical conception (which gives us infinite sets but not arbitrary collections) and a combinatorial conception (which gives us finite arbitrary collections but not infinite ones, unless we are going to countenance Bernays-style supertasks — and why, a Weyl might ask, suppose that such a fairy-story even makes sense?). Luca doesn’t take the occasion to comment, though: maybe he will return to question.

To be continued: Chapter 2 on the Iterative Conception next. But not until I’ve got answers to Exercises 41 online!

Luca Incurvati’s Conceptions of Set, 1

I’m really pleased to see that Luca Incurvati’s long-awaited Conceptions of Set and the Foundations of Mathematics has now been published by CUP. It’s currently jolly expensive. So let’s hope for an early inexpensive paperback. Happily, though, you will be able now to read an e-version of the book for free if your library has appropriate access to the Cambridge Core platform. So I’m going to assume I’m not the only one with access to the book! — and will dive in and comment slowly, chapter by chapter, over the next few weeks. I’ll be very interested (of course) to hear other readers’ reactions.

The first chapter is titled ‘Concepts and conceptions’. Not that Luca wants to suggest a sharp distinction here between concept and conception. But roughly, to characterize the concept of set is to characterize what someone has to grasp if they are to count as understanding ‘set’ (in the right way). But that characterization will leave open a lot of fundmental questions about the nature of sets and about what sorts of sets there are, about how sets relate to their members, and so on. And our answers to such questions will typically be guided by a conception of sets, which tells us something about what it is to be a set (a story which “someone could agree or disagree with though without being reasonably deemed not to possess the concept” of set). Take for example the iterative conception of sets: you don’t have to have grasped that surprisingly late arrival on the scene to count as understanding ‘set’. (I suppose that we might wonder about the understanding of someone who couldn’t see that the iterative conception, once presented, was at least a candidate for an appropriate conception of the world of sets: but reasoned rejection of the now standard conception would surely not debar you from counting as talking about sets.)

OK, so what does belong to the core concept of set as opposed to a more elaborated conception? Luca suggests three key elements:

  1. Unity: “A set is … a single object, over and above its members.”
  2. Unique decomposition: “A set has a unique decomposition” into its members.
  3. Extensionality: The familiar criterion of identity for sets — sets are identical if and only if they have the same members.

By the way, in talking about members here, it isn’t (I take it) being assumed that we we can call on a prior, fully articulated, notion of membership. The notions ‘set (of)’ and ‘member (of)’ have to be elucidated in tandem — just as e.g.  ‘fusion (of)’ and ‘part (of)’ have to be elucidated in tandem (and similarly for some other pairs).

These three aspects of the concept of set distinguish it from neighbouring ideas. (1) is needed to distinguish sets from mere pluralities — it distinguishes the set of Tom, Dick and Harry from those men. (2) is needed to distinguish sets from mereological fusions which can be carved into parts in arbitrarily many ways. (3) is needed to distinguish the relation between a set and its members, and the relation between an intensionally individuated property and the objects which have the property (different properties can have the same extension).

Let’s pause though over (1). We have two sets of Trollope’s Barchester Chronicles in the house. We can distinguish the two sets of six books, and count the sets — two sets, twelve individual books. One set is particularly beautifully produced, the other was a lucky find in an Oxfam bookshop. In a thin logical sense (if we can refer to Xs, count Xs, predicate properties of Xs, then Xs are objects in this thin sense) the sets can therefore be thought of as objects. But are the sets of books objects “over and above” the books themselves? Trying that thought out on Mrs Logic Matters, she firmly thought that talking of the set (singular) is just talking of the books (plural), and balked at the thought that the set was something over and above the matching books. Does that mean she doesn’t understand talk of a ‘set of books’?

The distinction we need here is the one made by Paul Finsler as early as 1926 in a lovely quote Luca gives:

It would surely be inconvenient if one always had to speak of many things in the plural; it is much more convenient to use the singular and speak of them as a class. […] A class of things is understood as being the things themselves, while the set which contains them as its elements is a single thing, in general distinct from the things comprising it. […] Thus a set is a genuine, individual entity. By contrast, a class is singular only by virtue of linguistic usage; in actuality, it almost always signifies a plurality.

In this sense, I’d say that (as with Mrs Logic Matters and the Trollopian sets) much ordinary set talk is surely class talk, is singular talk of pluralities. Luca cheerfully claims “if I say that the set of books on my table has two elements, you [as an English speaker] understand what I am saying”, I rather suspect that the non-mathematician, non-philosopher (i) is going to find the talk of ‘elements’ really rather peculiar, and (ii) is in any case not going to be thinking of the set as something over and above the two books, there on the table.

There’s little to be gained, however, in spending more time wondering how much set talk “as it occurs in everyday parlance” (as Luca puts it) really is set talk in Finsler’s sense, as characterized by Luca’s (1). I think it is probably less than Luca thinks. But be that as it may. Let’s move on to ask: what is the cash value of the claim that a set (the real thing, not a mere class) is a “genuine [bangs the table!] individual entity” [Finsler], is “a single object, over and above its members” [Luca]?

One key thought is surely that sets of objects are themselves objects in the sense that they too, the sets, can be collected together to form more sets. Suppose someone just hasn’t grasped that sets are the sorts of thing that themselves can straightforwardly be members of sets, would we say that they have fully cottoned on to the idea of sets (in the sense we want that contrasts with Finsler’s classes)?

Let’s take that thought more slowly. Suppose we for the moment take the idea of an object in the most colourless, all-embracing, way — just to mean a single item of some type or another. Then e.g. Fregean concepts are indeed items distinct from the objects that fall under them; fixing the world, there’s a unique answer to what falls under them; and they are individuated extensionally — same extension, same Fregean Begriff. This isn’t the place to assess Frege’s theory of concepts! The point, though, is that (1) talk of a single item distinct from the plurality it subsumes, plus (2) and (3), doesn’t distinguish sets from Fregean concepts. And similarly, I think, if we are to distinguish sets from (extensionally individuated) types in the sense of type theory.

But why should we distinguish Fregean concepts or types from sets? What, apart from some rhetoric and motivational chat is the real difference? Surely, one key difference is that Fregean concepts or types are, well, typed — only certain kinds of items are even candidates for falling under a given Fregean concept, or for inhabiting a given type. Sets are, by contrast, promiscuously formed. Take any assortment of objects, as different in type as you like — the number three, the set of complex fifth roots of one, the Eiffel Tower, Beethoven’s op.131 Quartet [whatever exactly that is!] — and then there is a set of just those things. At least, so the usual story goes.

Maybe that example is a step too whacky, and you could deny that there is such a set without being deemed not to know what sets are. But still, you’ll standardly want to countenance at least e.g. a set whose members are a basic set [either empty or with some urelement], a set with that set as it member, a set with those two sets as its members, etc. The set-forming operation does not discriminate the types of such things, but cheerfully bundles them together.

Isn’t the usual idea, in short, that a set of objects (objects apt for being collected into a set) is itself an object in the sense that it is, inter alia, apt to be collected into a set — indeed, collected alongside those very objects we started from? Whereas e.g. a Fregean concept has objects falling under it, but can’t be regarded as itself another item that could fall under a concept with those same objects — that offends against Frege’s type discipline.

I suppose — well, we’ll see when we come to his discussion of the iterative conception — that Luca could treat the idea of sets being (in a sense) type promiscuous as part of a certain conception of sets, something that elaborates rather than is part of our core concept. Given neither of us think there is a sharp concept/conception distinction to be drawn anyway, it certainly wouldn’t be worth getting into a fight about this. But my feeling remains that if we don’t say something more about how (1)’s understanding of sets as objects allows them to be themselves members of sets alongside other objects, then we won’t have done enough to distinguish the concept of set from the concept of more intrinsically typed items.

To be continued (with some comments on Chapter 1’s conception of ‘conceptions‘)

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