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‘)

12 thoughts on “Luca Incurvati’s Conceptions of Set, 1”

  1. Some thoughts:

    (a) Re “Does that mean she doesn’t understand talk of a ‘set of books’?” — It means she is using a different concept of ‘set’. When it comes to concepts and conceptions of sets, there’s no ‘Highlander rule’ (“there can be only one”).

    One of Incurvati’s aims in Ch 1, imo, is to explain what idea of sets he’s going to be talking about. On p 3, he mentions “the concept of set as it occurs in everyday parlance” and also “the concept of set as it occurs in elementary mathematics”. He doesn’t say they’re the same concept, or that the set concept he’ll be elucidating is meant to cover all, or even most, uses of ‘set’ in ordinary English, or that it’s necessary to understand it in order to understand ‘set’ in English.

    (b) When a set of books is the example, it’s natural for someone to think that the set isn’t an object over and above the books, because in that context talk of “objects” will seem to be about ordinary physical objects such as books. (And if you leave out “object”, what does “something over and above” even mean?) It can vary, though. Does a case of wine seem something over and above the bottles (and their contents)? Well, it might, if the bottles are in a physical box of some sort. If they aren’t, it might seem just a way to say “dozen”. (No one thinks a dozen is something over and above.)

    (If a child left their stamp collection at Grandma’s, is that the same as leaving the stamps there?)

    Once we start talking of abstract objects (or whatever we want to call the sorts of objects dealt with in mathematics), having a set be something in addition to its elements does not seem so strange.

    (c) If Incurvati’s 1-3 don’t distinguish sets from Fregean concepts or extensional types, perhaps they’re not meant to. His 1-3 are about the concept, and those things might be distinguished at the conception level instead (as the logical and combinatorial conceptions seem to be in §1.8).

    (d) What is supposed to be the problem with having things of different types in a collection? I don’t think it calls for being derided as “promiscuous”. Imagine a child having a collection that included stones, shells, leaves, and feathers. Who, unless they’re in the grip of some type-theoretic ideology, would say that’s not a collection? Even collections of things of the same type often have different types at the next level down. A book collection could include poetry along with fiction, history, etc, paperbacks, hardbacks. A mixed case of wine is still a case. If a problem appears when we talk of sets, rather than more generally of collections, what it is?

    (e) Absent urelements, what’s the “promiscuity” anyway? Everything’s a set.

    1. (a) Yes, Mrs LM is using something like Russell’s class-as-many as opposed to the concept of a set as something over and above its elements. Different notions, as I was saying. Different notions, as Luca says. However, on the set side, Luca does talk consistently about “the” concept of set. Yes, later he talks of ways we can sharpen up this concept. But at the outset he does write as if talk of a set (as opposed to class-as-many) of books and a set of numbers involves the same concept of set.

      (b) Well, that was my worry, about the cash value of “over and above”. You might be right that common-or-garden talk of sets of physical objects is more naturally construed as class-as-many talk, as against mathematicians talk of sets as things in addition to their elements. This relates to David Makinson’s observation.

      (c) I really doubt that Luca would want his initial characterization of the concept of set to allow Fregean concepts to count as sets!

      (d) There isn’t any problem about different types of things being in a collection — it was indeed my point that “set” seems accommodating like that! In particular, the usual story has it, any two sets can belong to another set. Contrast Fregean concepts. If they are at different levels of the hierarchy they can’t both fall under the same (monadic) concept.

      (e) OK, “promiscuity” was light-hearted (and misleading if it implied sinfulness!). We’ll see how the story develops re urelements: certainly, Luca allows them here at the beginning.

  2. I like the idea of trying to apply the concept/conception distinction here, so that it makes sense to talk about differing conceptions or theories of the same thing. Otherwise you end up with a kind of relativism – that ‘set’ means something different as you move from one theory to another.

    Just a thought about (1) “A set is … a single object, over and above its members.” I think there’s something to the idea that something like this essential to the (mathematical) concept of a set. However if we put the idea in (1) by saying “a set is not identical to any of its members” then what of ‘non-well-founded’ conceptions of set?

    Suppose someone thought there are sets which have themselves as members. Is that really now a different concept of ‘set’, so just changing the subject? Hence not a competing theory of sets but a theory about something other kind of thing?

    Suppose someone thought there is a set A whose only element is A itself. In what sense is the set A ‘something over and above its members’? It’s certainly not distinct from its members. Again, is such a conception really not a theory about ‘sets’ any more? Or could ‘set’ be a ‘family resemblance’ concept?

    If you want to allow self-membered things to still count as ‘sets’, what would be a better way to express (1)? Perhaps the core idea is that a set is something that can be a distinct object from its members? Is that what Finsler meant in the quotation by saying ‘a set is *in general* distinct from its members’? ie not always just the same thing as its members?

    1. Good point that saying that a set is an object “over and above” its members is naturally understood as ruling out self-membership. Though ruling out self-membership doesn’t rule out infinite downward chains that don’t circle back on themselves (and perhaps not even two-cycles!). So at least some questions about well-foundedness would seem to be left open by that characterization.

  3. David Makinson

    I suspect that in English the word ‘set’, in the rough sense of an arbitrary collection of possibly disparate objects, emerged from an initial mathematical usage, having been deliberately introduced into the language of English-speaking mathematicians in the early 20th century when translating Cantor or other texts expounding his new discipline. I remember that when I first encountered the word in that sense as an undergraduate in Sydney in the early 1960s, I found it quite strange. I knew it as designating a collection of similar objects that are used together for a specific purpose, e.g. a set of clothes, a set of dishes. Outside those contexts it hardly seemed to occur. It would be interesting to find the first occurrences of this word in the English-language mathematical literature (perhaps replacing ‘manifold’), philosophical literature (presumably replacing ‘class’, ‘collection’) and in common parlance (in a sense broader than that for cutlery). Of course, nothing philosophical hangs on this, but it could help with perspective. A nice job for student of the English language!

    1. I suspect you are right — about the use of set for arbitrary collections going from mathematical English to common parlance, rather than the opposite direction. And it still strikes me as odd to talk as Luca does about e.g. the set of books currently on my desk [a pretty arbitrary collection!] as oppposed to e.g. my set of the works of Jane Austen. I’m not sure I’d even talk of my set of Wittgenstein’s works, as (i) what to count, and how complete is it? and (ii) it’s a scruffy lot, from different publishers.

      1. Why does any of that matter? There’s a different notion of sets that’s normally used in English in cases like your Austin one, and isn’t used for ‘arbitrary’ collections. (‘Collection’ isn’t normally used then either.) You seem to see this as somehow casting doubt on the mathematical notion, but why? What, if anything, is the actual problem?

        Fans of plural logic often argue that many cases are more naturally treated as plurals in English than as sets / collections / etc, and they’re right up to a point, but they end up talking in some quite odd ways too.

        BTW, the things commonly brought up as supposedly odd, weird, or nonsensical about set theory sets — the empty set, sets of 1 element, sets containing different types of things — also occur in programming languages without anyone having much trouble understanding them.

        I’ve taught programming in several different language to people from a wide range of backgrounds (and have also seen what happens when other people were teaching), and while there are some things that people struggle to get their heads around, empty, 1-element, and mixed lists, sets and other sorts of collections aren’t among them.

        1. I don’t think it matters a lot (any more than David does). Except that people can and do appeal to a supposed prior non-mathematical sense of “set” that we are all supposed to understand in allegedly explaining the mathematical usage. And maybe our common-or-garden usage helps less than they like to think!

          1. I think there pretty clearly is a prior, non-mathematical sense of ‘set’, ‘collection’, ‘case’, ‘flock’, and other terms that treat multiple things as one, as well as some verbs such as ‘collect’, that does help in explaining and understanding mathematical sets. That’s one reason why when people are given an explanation that begins “a set is a collection”, and some examples, they don’t start saying “what?”, “how?”, “that makes no sense!”

            It might not seem that way if you focus specifically on ‘set’ and on uses that least fit the mathematical concept, or if you think there’s something suspect about set theory sets so that their legitimacy depends on exactly matching some other usage in the very areas considered problematic. (“And maybe our common-or-garden usage helps less than they like to think!” — How much help does it need?)

            Even with ‘set’, however, there are examples that are closer to an ‘arbitrary collection’. Consider train sets. I looked ‘train set’ up just now, and one example was a John Lewis wooden train set that contained models of many things: an engine, cars, and tracks, of course, but also road signs, people, buildings, trees, and farm animals. And it doesn’t have to stop there. A train set could include wild animals, a mountain with a tunnel, perhaps even a pond or lake, complete with ducks. There doesn’t seem to be a hard limit. A fantasy train set could include giants and a dragon. (That’s before we even get to imaginary train sets that might appear in fantasy or science fiction.)

            Of course it can be pointed out that the ‘elements’ are all models made of wood, metal, and plastic, and that they’re (usually) used together for a specific purpose. Similarly, the sets used in mathematics will have elements that are all mathematical objects (usually) brought together for some purpose. It’s just not built into the definition.

  4. Luca Incurvati

    Thank you for going through my book and for the comments.

    A few clarifications/observations:

    1) My claim is precisely that when people talk about the set of books on the table, they don’t necessarily mean set as a unity. See p. 3: ‘when I said that the set of books on my table has two elements, it would have been prima facie legitimate for someone to take me simply to be saying that there are two books on my table’. So the initial point of that section is that whilst some ordinary set talk can indeed be interpreted as class talk (in Finsler’s sense), we’re going to be interested in the concept of sets as unities. So the gist of what’s going on here is along the lines of what Rowsety Moid said.

    (By the way, not that it matters for my purposes, but I take it it is established in historical linguistics that the usage of ‘set’ in English as a collection of things predates the mathematical usage. See, e.g., https://www.etymonline.com/word/set .)

    2) About typed entities: that’s why I used ‘object’ in Unity rather than ‘item’. Replacing ‘object’ with ‘item’ leaves out an important aspect of Unity, which is precisely the one that serves to ensure that sets are the kinds of things that can themselves be collected together. Also, note that after laying down Unity I say: ‘In particular, a set is a single object bearing a certain distinguished relation [the containment relation] to the objects a class as many aa comprises’.

    I suppose I just don’t see why we need to say anything more than that sets are objects to ensure that they are the kind of things that *may* be collected into a set. If you want to say: ‘Oh, and sets can themselves be members of other sets’, one could reasonably ask: ‘And why?’, and the best answer here is surely: ‘Because they are themselves objects’. One could of course insist that this is still compatible with sets *as a matter of fact* not being members of other sets. But the point is that any particular principle about which sets of sets there are is likely to be one that is part of a particular conception. Consider, for instance, someone who thinks that there are only finite sets of Urelemente (see e.g. the Kreisel quote on p. 73, fn. 4). Do we really want to say that such a person doesn’t grasp the concept of set?

    Also, one could insist that taking sets to be objects is still compatible with sets not being members of other sets because of typing. Let’s consider a couple of possibilities. Take first, as a typed conception of collection, that of a Fregean concept. Taking sets to be objects rules out Fregean concepts as sets because Fregean concepts (if Frege is right) are not objects. (In addition, what I say after Unity in the remark above also rules out Fregean concepts as members of sets.) Things are different if concepts may be objects. But I think that’s how it should be. In particular, consider another typed conception of collection, the one given by the class-theoretic interpretation of type theory, where we have individuals at type 0, classes at type 1, classes of classes at type 2, and so on. These typed entities do satisfy Unity, Uniqueness of Decomposition and Extensionality. But this doesn’t seem a bad result to me. Do we really want to rule out, against Russell and several others, classes in this sense as candidates for sethood?

    3) Having said all of this, I am not committed to the claim that Unity, Uniqueness of Decomposition and Extensionality are sufficient to single out the concept of set. The point is to get to the concept we’re going to be interested in.

    1. Many thanks for this! Very helpful, of course.

      (2) is the important point here, that you are leaning on “object” (as opposed to e.g. “item”) to do a significant bit of work.

      Which is what my remarks were trying to bring out, in a probably too laboured way! We are on the same page on this: we need to construe the talk of sets as objects as intended to distinguish them from typed entities like Fregean concepts (and then that opens up the possibility of sets being members of other sets).

      Re-reading my remarks, though, I think I make it sound as if I thought that there were two steps here, i.e. treating sets as [Fregean] objects, and treating sets as objects “apt to be collected into a set”. I don’t think I meant to imply that!

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