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 C 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!
7 thoughts on “Luca Incurvati’s Conceptions of Set, 2”
Thanks again Luca!
(1) Yes, as I said, it may turn out that one sharpening is particularly fruitful, but of course it may turn we have more than one nice option. (We can quibble about how best to describe what the self-aware sharpener is doing, either way, but I agree that it isn’t very important, as long as we are clear what’s going on!).
(3) When I said “‘If we don’t take e.g. supertasks seriously, then just what is the content of the combinatorial conception?’” I didn’t mean to imply we should take supertasks seriously! The point was just that — assuming we don’t take Bernay’s talk of infinitely many independent acts literally — we need some elucidatory story to give substance to the combinatorial conception. As I think you agree. (I suspected, perhaps quite wrongly, that R.M. was too quick to think we had a grasp of the combinatorial conception, without such a story in hand.)
Thank you for the further comments.
1) About conceptions-as-sharpenings: a conception is a possible answer to the question ‘What is an F?’ that someone could agree or disagree with without being deemed not to possess F. It need not be the only one, although it can be, and it can certainly be the best one and hence we may want to actually sharpen the concept of F accordingly. That is linked with my view that what’s part of the concept and part of a conception is not fixed: what was at some time only part of a particular conception may at a later time become part of the concept.
So my characterization of a conception is compatible with the sharpener thinking there is more than one possible way of sharpening the concept, although they might also think one is going to be the best one because it satisfies various desiderata — see the method of inference to the best conception later on in the book. Take again the case of computability. To be effectively computable is an answer to the question ‘What is it to be computable?’ that someone could agree or disagree with without being deemed not to posses the concept of computability. It is not the only possible answer (see feasible computability), although it can be the best one because of, e.g., its mathematical fruitfulness, and so might actually want to sharpen the concept of computability in that direction.
In short, it seems to me that being a sharpener about the concept of F implies neither pluralism nor singularism (but not of the conception-as-analysis type, of course!) about F.
2) About conceptions-as-replacements: on this approach, we are officially not talking about conceptions of, say, set but about conceptions of set* (where sets* can serve much of the practical or theoretical role that sets were meant to serve). That’s indeed the case, but it still fits with my account of what a conception is: after all, a conception of set* is still a possible answer to the question ‘What is a set*?’ that someone could agree or disagree with without being deemed not to possess the concept of set*. Of course, we need to say something about what needs to be in set*. That will have to do with what features of set we want to preserve in order for set* to play a similar practical or theoretical role.
3) In response to Rowsety Moid you say: ‘If we don’t take e.g. supertasks seriously, then just what is the content of the combinatorial conception?’. One possible answer would be: that membership in a collection is primitive and not determined by whether the membership candidate satisfies a certain condition or has a certain property (or some such). This doesn’t mean that membership in a collection is determined by supertasks or infinite acts or anything of the sort. Indeed, in your third post on my book you quote me as saying that according to the combinatorial conception ‘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’. This accords with what I take to be the central component of the combinatorial conception, namely the rejection of constraints on collection existence due to e.g. definability or expressibility. Of course, this doesn’t mean that a combinatorial conception has an easy time dealing with the Axiom of Infinity. I agree with that. But it’s not clear that a logical conception will always have an easy time of it either: the issue is that as soon as you draw some restrictions on which properties determine a collection or which conditions determine a property (which you must do to avoid paradox), these restrictions might also prevent you from getting Infinity without additional stipulations. Two cases of this sort are certain ways of restricting Comprehension based on limitation of size (which I discuss in Ch. 5) and stratification (which I discuss in Ch. 6; here I’m thinking in particular of NFU, which, unlike NF, doesn’t deliver Infinity, although it is consistent with it).
(b1 & b2 (b3)) There’s an ancient objection to completed infinities. (I see your Quineans and raise you Aristotelians.) My point was that it’s not enough to have a test that can be applied to any object and tell you whether it’s a natural number or not: it must also be possible to form a completed infinite set of the ones where the answer is “yes”. The logical conception shouldn’t get that for free.
Put another way, why does the combinatorial conception get stuck with a supertask while the logical conception is allowed a leap to an infinite set without any effort? If we’re going to think in terms of supertasks, the logical conception has one too.
Comments on comments on comments on comments are subject to the law of diminishing fleas, so I’ll be very brief.
(b1) Logical conception. It there is anything in this at all, it surely has to be the idea that — barring ‘nasty’ cases — the objects that fall under a property form an extension which is a set. Sure, we have to say something about nastiness, to rule out the paradox-engendering cases. But if it remains, on this conception, an open question whether even “is a natural number” is nasty (i.e. has a set as extension), then the supposed conception would seem to have no content. (So I’m pretty sure that when Luca talks about a logical conception he means a conception which settles more or less immediately that there is an infinite set of natural numbers!)
(b2) OK, you want to take talk of infinitely many independent acts, talk of supertasks etc. as an image/metaphor that needs to be cashed out. Me too! If we don’t take e.g. supertasks seriously, then just what is the content of the combinatorial conception?
(b3) Platonism, generally. There’s plentitudinous [in for a penny, in for a pound; if you buy numbers, buy all the sets you can dream up!] platonism: any consistently postulated abstract objects exist. But there’s also parsimonious platonism (Quine): only buy abstract objects when you have to [have to for what? — a good question, for the Quinean]. Not the place to have this debate. But a Quinean will want reasons for taking each new step.
(a) You seem to see the way replacement, sharpening, and analysis fit together a bit differently than Incurvati, though I’m not sure how significant this is.
You have replacement and then a sharpening – analysis spectrum. Incurvati sees analysis and replacement as two alternatives to sharpening, says he will typically use sharpening, and adds “what we will be saying can easily be rephrased in terms of the other two approaches.” (p 21)
(b) The rhetoric in the second half of your final paragraph — conceived in sin, unholy alliance, fairy-story — suggests an ideologically- or even ‘religiously’-inspired antipathy to standard set theory, rather than well-reasoned philosophical or mathematical objections. I also feel an urge to add, Wikipedia-style, ‘citation needed’.
What’s the ‘sin’? Why is the ‘alliance’ ‘unholy’? What shows that the logical conception can really give us infinite sets (without assuming their existence via an axiom of infinity), or that the combinatorial conception can’t, or that the image of infinitely many independent acts should be taken so literally? (Even a large finite number of choices could not actually be carried out.)
(a) Not that significant. Though I don’t think that the options carve as neatly as Luca’s presentation suggests.
(b) “Conceived in sin, unholy alliance, fairy-story” were supposed to be light-hearted. The jest fell flat. I was alluding to one strand of anti-Platonist worry (in more recent times e.g. explored by Michael Dummett and Crispin Wright, but going back a hundred years). And I was wondering aloud if Luca was going to mention it. [As to the particular remarks: the logical conception surely gives us infinite sets like the set of natural numbers, since “is a natural number” is a kosher property in anyone’s book. And there is surely a conceptual difference we can/should between the idea of the result of a finite process — even if one we can’t carry out for reasons of time and space — and the result of an infinite process (compare effective computability with hypercomputability).]
(a) I think a potentially interesting question is whether what Luca says in sharpening terms really can “easily be rephrased in terms of the other two approaches.”
(b) I’m happy for it to be light-hearted; I still think the terms in which it was expressed are significant and fit a certain type of antipathy to standard set theory that seems to be behind (for example) some of the objections made by some advocates of category theory or plural logic.
Re the logical conception giving us infinite sets, it’s not enough that “is a natural number” is a kosher property: it in addition has to be possible to form a completed infinite set of them. At least that’s how it seems to me. And if that is possible, why can’t the same set be seen combinatorially? (In a programming analogy, a program can carry out a sequence of computations and collect the results in a set. That set then exists and has whatever elements it has. How the elements were produced no longer matters.)
Or, taking a different route, if it’s possible to apply “is a natural number” an infinite number of times, why can’t there be an infinite number of choices or membership decisions to build the combinatorial set that way?
Your ‘supertask’ argument seems a strange one anyway. It starts by saying the combinatorial conception “can be associated with images such as a sequence of random choices” — can be, associated, images — which doesn’t look like a necessary connection. Next Bernays is quoted on “infinitely many independent acts”, with nothing that says we have to take that literally in any way. Then there’s a leap to saying the combinatorial conception does not give us infinite collections “unless we are going to countenance Bernays-style supertasks”. How did images that merely “can be associated” turn into something necessary?
Re Platonism, I can understand rejecting mathematical Platonism so that no mathematical objects exist. It can seem quite strange to think that numbers, for example, really exist. What does that even mean? What sort of existence is it? But if someone thinks some mathematical objects exist, they’ve already taken the difficult step; and if numbers exist, why not sets of them, even infinite sets?