I’ve recently run into two arguments that there is no single intuition behind the notion of set. One was a talk by Tatiana Arrigoni in Berlin that I mentioned earlier, arguing that there may be multiple intuitions underlying set theory. Another was a talk by Jose Ferreiros at the Berkeley Logic Colloquium last week, arguing that there isn’t even *one* intuition underlying the notion of set.

Ferreiros started by pointing out that the 19th century founders of modern set theory (Cantor, Frege, and Dedekind) all seemed to be working with different notions. In particular, Cantor seemed to never consider a set as an element of a further set (his famous theorem was apparently proved about the set of functions on a given set, rather than using the modern notion of a power set). In addition, recognizing some paradoxes that would result from this theorem, he rejected the notion of a set of everything. His notion was fairly close to the intuitive notion of “collection” that we often introduce students to set theory with.

Frege (and Russell, following him) thought of sets as extensions of concepts, or as some way of dividing the universe into two parts. It’s a more “top-down” notion than the others, because of the sort of impredicativity involved in dividing a totality (including the set to be defined) into two parts. This notion is probably closest to our intuitive notion of “property” that the set concept seems to inherit some of from the separation axiom.

Dedekind had a slightly different conception from either. I didn’t quite get the details, but it sounds like it was the only notion that was relatively close to the modern “iterative conception” championed later by Gödel (among others).

The fact that our modern notion of set has inherited properties from each of these suggests that it is not a rigorization of any of these intuitive concepts. He also gave some other arguments, based on the fact that we have intuitions about the existence of “absolutely arbitrary” subsets of any given set, which aren’t successfully cashed out in any of our axioms, but are approximated by the axioms of separation and choice.

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Michael Greinecker(07:28:23) :What’s wrong with the cumulative hierarchy conception? It certainly agrees with the theory and has a strong intuitive foundation.

Michael Greinecker(07:51:42) :“The fact that our modern notion of set has inherited properties from each of these suggests that it is not a rigorization of any of these intuitive concepts.”

But we still see it as a formalization of the iterative conception. So what is the problem?

Kenny(11:36:23) :The problem is just that we also see it as a formalization of the notion of “collection”, which

doesn’tseem to have any sort of iterativity to it. And people don’t generally grasp the iterative conception until they’ve already studied some set theory somewhat formally.Aidan(13:07:25) :There’s also Boolos’ point at the end of ‘Iteration Again’ that neither the iterative conception nor limitation of size grounds all of the axioms of ZFC (If my memory serves me rightly, IC doesn’t seem to imply extensionality, and much more worryingly, choice or replacement. LOS doesn’t imply powerset or infinity, and actually implies ~union). So if the appeal to some underlying pre-axiomatic conception of set is supposed to bestow special status on the axioms of ZFC, it looks like neither of the standard ones will do the trick alone.

Michael Greinecker(02:08:05) :I agree that it is hard to get a intuitive noton before being familiar with axiomatic set theory, but that`s probably true in most abstract branches of mathematics.

Extensionality simply says “what” one builds the IC upon, so I see no problem with intuition here. Choice comes out of the fact that “all” sets existing at a stage exist.

Replacement is harder to motivate, but I remember reading a iterative justification in some textbook.

Aidan(09:31:01) :Extensionality is unproblematic, I agree. Boolos does offer iterative justifications for both choice and replacement himself, but his claim was they require strengthenings of the stage theory that build in content that wasn’t there in our original conception.

And I’m not seeing how the proposed justification of choice goes. That “all” sets existing at a stage exist looks like a tautology, so I’m guessing you wanted to say something like that at each stage we form “all” arbitrary sets of things existing at the earlier stage. As Hallett points out (1984: 218-9), what “all” amounts to here depends on how strong a version of restricted comprehension you build into your framework. Even building in full classical restricted comprehension into your framework (as Boolos does with his ‘specification axioms’), I don’t see how we’re going to get a justification of choice here. We can’t just assume that every choice set exists at some level, since the original stage theory doesn’t contain any set-forming machinery strong enough to guarantee that.

Aidan(09:40:47) :“I’m guessing you wanted to say something like that at each stage we form “all” arbitrary sets of things existing at the earlier stage”.

The sharp-witted amongst you will notice that this is insensitive to what Kenny wrote in his post about how our axioms allow us to at best approximate here. Let’s just be sloppier, and say at a given stage, all possible sets of objects formed by the previous stage have been formed (which is pretty much how Boolos puts it in his informal gloss). That’s totally neutral on whether we can form absolutely arbitrary subsets. The point in my previous comment, then, is just that “all possible sets” is constrained by what set-formation principles you’ve allowed yourself; Boolos’ point is that we won’t get choice out without putting something choice-y in.

Michael Greinecker(10:02:38) :That’s why I’ve put “all” in scare-quotes. I think a reasonable interpretation is “as many as possible”.

“Boolos’ point is that we won’t get choice out without putting something choice-y in.”

That is pretty obvious, but the question is wether being choice-y fits well within our intuitive notion of the IC. I think so. Does the IC give any reason to suppose choice should fails somewhere? I doubt it.

Aidan(10:12:05) :“Does the IC give any reason to suppose choice should fails somewhere? I doubt it.”

But that wasn’t the point – there was never any hint of a claim that IC might imply or support or even just suggest *failures* of choice.

The project was to codify our conception of set (in the case of IC, in the stage theory; in the case of LOS, in NewV), and to justify the adoption of axioms based on the fact that they followed from that codification. The worry is that neither conception (suitably codified) implies all of the axioms of ZFC, so neither justifies all of those axioms.

(There are, of course, a number of alternative ways to approach the epistemology of set theoretic axioms. I’m aware the point I’ve been making hinges on this Boolos/Potter (and Goedel??) way of thinking about things.)

Kenny(10:17:47) :Also, there’s the question of whether ZFC is the proper way to characterize any of these intuitions. If Choice is the only part getting us arbitrary sets, then it seems like we need more. Large cardinals extend the length of the whole series, and some of them might also be seen as increasing the width (for instance, measurable cardinals prevent V=L, so they must have some sort of width effect). But it’s really not clear that they’re sufficient that way, on each of these conceptions.

However, they do seem to be well-justified, which seems to me to make this all the worse for the intuitive conceptions of set.

Michael Greinecker(10:22:19) :I’m not sure I agree. I would say that IC and limitation of size are two very different views on the topic, so it’s hard to bring them together into a coherent intuition of the universe of sets.

Introducing a choice-y view on set generation with an IC leads to a very coherent intuitive view of the universe of sets on the other hand. There may be different versions of the IC, but it is perfectly okay to single out the choice-y by a certain intuition on set generation that is taken over all stages.

Aidan McGlynn(10:44:44) :Michael,

I actually agree (in fact, I wrote a paper while ago, in which I invest no confidence, trying to chart just how hard it was to get IC and LOC to mesh in the way necessary). I think it’s a far better option to try to build plausible stuff into IC that will get us choice and replacement.

So in my original post, I was merely putting the Boolos thought forward as another reason for thinking our conception of set, if we have such a thing, is fragmented. I’m pretty reluctant myself to draw that conclusion.

John Ryskamp(16:12:59) :The problem is that the notion of a set is too vague an assumption to do anything with. The closest anyone came to making it more precise was Cantor, and that was a disaster. I don’t know if your readers are familiar with Garciadiego’s book on Russell and the set-theoretic paradoxes, but in spite of the endless typos in that book, I strongly recommend it. Read the footnotes closely too.

Natural mathematics–designed to “avoid” or “solve” non-existent “paradoxes:–has had a huge and pernicious effect, one which I trace in this paper:

Ryskamp, John Henry, “The Unity of Twentieth Century Ideas” (April 14, 2006). Available at SSRN: http://ssrn.com/abstract=897085

Enigman(09:35:24) :But we can begin with the notion of a collection (without the overtones of any acts of collecting) or a plurality (including the degenerate case of an individual), which is surely one of our most precise notions (for all that we find picking a name for it tricky, these days). There seems nothing imprecise or vague about there being some particular number of different and definite things.

The problem, as I see it, with sets is that we inevitably get proper classes with them. So on the one hand we want sets to capture our intuitive notion of a collection, but on the other hand a proper class is intuitively a collection (of sets). So, we can define various kinds of sets, to do various things, but they cannot capture the fundamental notion (not even en masse, as they will form a proper class).

John Ryskamp(10:18:38) :The problem is not “beginning” with a “notion,” or for that matter, with “intuition.” The problem is what internally consistent role those ideas play in any argument. Cantor’s ideas simply have no logical content. If you don’t want your argument to have any logical content, then by all means use them. But don’t expect–as you do–people to “grant” that your arguments don’t make any sense because, somehow outside of internally consistency, we must acknowledge “beginning” or “notions” or “intuition.” Didn’t anyone ever teach you about rigor in argumentation.

Anyway, just read Garciadiego. You are far from coming to terms with the real issues in set theory. You’re just very ignorant. Easily cured: study.