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!