Thursday, January 18, 2007
Let's talk about sets, baby
A while ago, I wrote a paper exploring the late great George Boolos' suggestion that our pre-axiomatic conception of set might be a combination of the standard iterative conception and the limitation of size conception. Boolos' project, as I understand it, was to try to justify the axioms of ZFC by codifying these pre-axiomatic conceptions (into the stage theory and New V respectively), then proving the axioms of ZFC from the codifications. Or that's the shape Boolos' project would have taken ideally; rather he was trying to show the limits of this kind of approach by showing that neither IC or LOS could justify all of the axioms of ZFC in this manner. IC failed to imply extensionality, choice and replacement, while LOS failed to imply power set and infinity, and in fact implied ~union. The suggestion, I take it, was that some hybrid conception of set formed out of IC and LOS might imply all of the axioms.
There are a couple of immediate obstacles to be overcome. First of all, IC implies union, while LOS implies ~union. Secondly, take a collection t such that t = {t}. t isn't part of the iterative hierarchy, and so isn't a set by the lights of IC, but t obviously is a set if your characterization of set-hood is simply being of sufficiently small cardinality, as it will be if one adopts LOS. I argued that we should be very pessimistic about the prospects for solving these problems in a manner that allowed for the success of the general project; that of justifying the axioms of ZFC with appeal to something recognizable as (a codification of) our pre-axiomatic conception of set.
There are a number of responses one might have. One might just dismiss this as a wrong-headed way of approaching the epistemology of set theory. Or one might wonder why the fixation with ZFC (see here for some discussion of related issues). These are important questions, but my project was just to see how far such an epistemology of ZFC could fly.
In conversation, my friend Nikolaj Pedersen suggested that I'd simply overstated the difficulties in formulating the hybrid conception of set. Sadly, he never gave me any idea of what he had in mind. Do any of the more mathematically minded among you have any idea what kind of strategy might work here? (I could just email Nikolaj, but where's the fun in that?)
There are a couple of immediate obstacles to be overcome. First of all, IC implies union, while LOS implies ~union. Secondly, take a collection t such that t = {t}. t isn't part of the iterative hierarchy, and so isn't a set by the lights of IC, but t obviously is a set if your characterization of set-hood is simply being of sufficiently small cardinality, as it will be if one adopts LOS. I argued that we should be very pessimistic about the prospects for solving these problems in a manner that allowed for the success of the general project; that of justifying the axioms of ZFC with appeal to something recognizable as (a codification of) our pre-axiomatic conception of set.
There are a number of responses one might have. One might just dismiss this as a wrong-headed way of approaching the epistemology of set theory. Or one might wonder why the fixation with ZFC (see here for some discussion of related issues). These are important questions, but my project was just to see how far such an epistemology of ZFC could fly.
In conversation, my friend Nikolaj Pedersen suggested that I'd simply overstated the difficulties in formulating the hybrid conception of set. Sadly, he never gave me any idea of what he had in mind. Do any of the more mathematically minded among you have any idea what kind of strategy might work here? (I could just email Nikolaj, but where's the fun in that?)
Labels: Philosophy of Mathematics, Set Theory
# posted by Aidan @ 10:56 PM 
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First of all, IC implies union, while LOS implies ~union.
Is this really clear? LOS doesn't obviously imply ~union, depending on how you take it. If you take it to be limitation on size of transitive closure, then union doesn't cause any troubles. And even if it's the size of the set itself, then union (by itself) can only introduce a single new cardinality, which is just the limit of the cardinalities you already have available.
So it's not totally clear that union is incompatible with LOS.
Secondly, take a collection t such that t = {t}. t isn't part of the iterative hierarchy, and so isn't a set by the lights of IC, but t obviously is a set if your characterization of set-hood is simply being of sufficiently small cardinality, as it will be if one adopts LOS.
However, set-hood requires not just bing of small cardinality, but also existing. So if one has other metaphysical reasons to reject non-well-founded collections, then those will go through to reject considering them as sets (since they don't exist).
But yeah, this is all tricky stuff.
(Coincidentally, the last two letters on the CAPTCHA I had to fill out for this comment are "zf".)
Is this really clear? LOS doesn't obviously imply ~union, depending on how you take it. If you take it to be limitation on size of transitive closure, then union doesn't cause any troubles. And even if it's the size of the set itself, then union (by itself) can only introduce a single new cardinality, which is just the limit of the cardinalities you already have available.
So it's not totally clear that union is incompatible with LOS.
Secondly, take a collection t such that t = {t}. t isn't part of the iterative hierarchy, and so isn't a set by the lights of IC, but t obviously is a set if your characterization of set-hood is simply being of sufficiently small cardinality, as it will be if one adopts LOS.
However, set-hood requires not just bing of small cardinality, but also existing. So if one has other metaphysical reasons to reject non-well-founded collections, then those will go through to reject considering them as sets (since they don't exist).
But yeah, this is all tricky stuff.
(Coincidentally, the last two letters on the CAPTCHA I had to fill out for this comment are "zf".)
Thanks Kenny, I was really hoping to get feedback from you. It's too late for me to venture a proper response tonight (nor am I even sure I'll be able to give one soon), but let me at least give the standard Boolos argument for LOS -> ~union. Let t be the singleton of V. t itself is set-sized, but the collection of the members of its member trivially isn't (Boolos is taking BIG to mean bijectable with V).
It's true that the argument isn't watertight (there are some resistible assumptions in there about the kinds of memberships relations proper classes can enter into, for example). In general though, I've tended to take Boolos' results at face-value, since he was an infinitely better logician than I am. That's part of the reason I'm looking for feedback from other people way more technically competent than me.
It's true that the argument isn't watertight (there are some resistible assumptions in there about the kinds of memberships relations proper classes can enter into, for example). In general though, I've tended to take Boolos' results at face-value, since he was an infinitely better logician than I am. That's part of the reason I'm looking for feedback from other people way more technically competent than me.
Actually, I should write this while it's fresh in my mind. Part of my paper tried to argue that as soon as we mess with the notion of set-hood in the way you suggest, we can no longer justify all of the axioms we wanted to.
The weak limitation of size principle can be taken here as; objects form a set if they are in one-one correspondence with the members of a given set. It's this principle that provides the cute justification of replacement one finds in Hallett's book, and in Potter and Burgess' most recent books. Am I wrong in thinking we'll lose this principle if, for example, we reject non-well-founded sets?
(I also argued that Boolos' much less direct argument that LOS justifies replacement fails on this proposal. Of course this doesn't enable me to conclude that there's no way to pull the trick, but I did think it merited the kind of pessimism I mentioned in the post. One can see the dialectical work done by insisting that we can all of the axioms of ZFC here, but I'm not going to apologize for that - I've already bitten off more than I can chew.)
The weak limitation of size principle can be taken here as; objects form a set if they are in one-one correspondence with the members of a given set. It's this principle that provides the cute justification of replacement one finds in Hallett's book, and in Potter and Burgess' most recent books. Am I wrong in thinking we'll lose this principle if, for example, we reject non-well-founded sets?
(I also argued that Boolos' much less direct argument that LOS justifies replacement fails on this proposal. Of course this doesn't enable me to conclude that there's no way to pull the trick, but I did think it merited the kind of pessimism I mentioned in the post. One can see the dialectical work done by insisting that we can all of the axioms of ZFC here, but I'm not going to apologize for that - I've already bitten off more than I can chew.)
'The doctrine comes in at least two versions: On a strong version of limitation of size, objects form a set if and only if they are not in one-one correspondence with all the objects there are. On a weaker, there is no set whose members are in one-one correspondence, but objects do form a set if they are in one-one correspondence with the members of a given set. (Under certain natural conditions, this last hypothesis can be weakened to: if there are no more of them than there are members of a given set.)'
(Logic, Logic, and Logic: 90)
I think I need to retract what I wrote in my last comment. What you no longer get is the weakened weak limitation of size principle. You should still get the original weak limitation of size principle, which justifies replacement. Urg.
(Logic, Logic, and Logic: 90)
I think I need to retract what I wrote in my last comment. What you no longer get is the weakened weak limitation of size principle. You should still get the original weak limitation of size principle, which justifies replacement. Urg.
To me it seems that the deeper thing surrounding these issues is the more general ontological question. You suggest, "objects form a set if they are in one-one correspondence with the members of a given set." But the counterexample you and Boolos give is V (and no other objects). However, this only works if V is itself taken to be some sort of object. (It's not clear why it would or wouldn't be.)
Similarly for non-well-founded sets. While these are small enough to be sets, it's just not clear that the elements of one of these sets are really all objects (especially since in many cases, one of those elements is the putative set itself whose existence is in question).
Limitation of size alone isn't going to ban non-well-founded sets, though I could see what an iterative conception might.
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Similarly for non-well-founded sets. While these are small enough to be sets, it's just not clear that the elements of one of these sets are really all objects (especially since in many cases, one of those elements is the putative set itself whose existence is in question).
Limitation of size alone isn't going to ban non-well-founded sets, though I could see what an iterative conception might.
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