the boundaries of language

I'm off to Buffalo this weekend for Reason, Intuition, Objects: The Epistemology and Ontology of Logic, where I'll be presenting my paper 'Iterations and Limitations'.

For the interested, here's the abstract:

'The iterative conception of set has been defended as a natural and non-arbitrary successor to the inconsistent naive conception, but in ‘The Iterative Conception of Set’ George Boolos showed that the hierarchical picture of the set-theoretic universe given to us by this conception fails to lend support to some of the axioms of ZFC, most notably choice and replacement. Both these axioms are delivered by a rival conception of set—the limitation of size conception—but unhappily this puts the axioms of power set and infinity beyond our reach, and has struck many as merely a technical device designed to avoid the paradoxes, rather than a genuine elucidation of our conception of set. Boolos has suggested that perhaps our conception of set is a hybrid of the leading thoughts behind the iterative conception and limitation of size, and in this paper I begin an assessment of the prospects of such a conception. I argue that even if this hybrid conception—the limitation of iteration conception, as I call it—can deliver all of the axioms of ZFC, it does so only if we are willing to make assumptions justified (if at all) only on pragmatic grounds. Insofar as our project is that of providing conceptual grounds on which to believe the axioms of ZFC, I conclude that we have reason to reject the limitation of iteration conception.'

Labels: Conferences, Set Theory

I've been having a discussion with Joe Salerno over at his blog Knowability on whether moving away from the standard Lewis-style semantics for counterfactuals with impossible antecedents (as he and Brit proposed in their recent piece in the Reasoner) might allow us to respond to some worries Michael Potter had in his most recent book concerning a particular way of thinking of the priority of members over sets in platonist interpretations of the Iterative Conception of set. (Phew!)

I'm not sure whether the suggestions will pan out at all, but in case some of you might be interested, Joe's post and our discussion are here.

Labels: Links, Set Theory

I've just finished reading Alex Paseau's 'Boolos on the Justification of Set Theory', which I really liked. Actually, I didn't just like it - I thought it was pretty much right. But this passage, discussing the relative merits of the iterative conception and limitation of size, brought up something I've been puzzling over for longer than I care to admit:

'In the second-order setting here discussed, Limitation of Size (either Cantor or von Neumnan--the two are equivalent if the universe is well-orderable) is a consequence of the iterative conception. The iterative conception's explanation and justification of Limitation of Size is that 'large' collections do not constitute sets because there is no stage at which all their elements are available for formation. Limitation of Size is thus a natural consequence of the iterative picture. The alternative of taking it as primitive, be it in a Fregean context or otherwise, seems wrong-headed.' (50)

von Neumman Limitation of Size is:

some things form a set iff they are not in one-one correspondence with the universe (47),

while Cantor Limitation of Size is:

some things form a set iff they are not in one-one correspondence with the ordinals (48).

I just don't see how these principles are even consistent with the iterative conception, let alone consequences of it. The left to right directions of each look like straightforward consequences of the iterative conception, for just the reason Paseau offers in the text. But the right to left directions just look false given the iterative conception's commitment to well-foundedness. Let x be a collection such that x = {x}. x is not in one-one correspondence with either the universe or with the ordinals, having only one element. But since V = WF, it ain't a set either.

Am I missing something obvious here again?

Labels: Set Theory

I wanna read this article. Sadly, UT doesn't subscribe to Philosophia Mathematica, which really is such a shame. Sigh.

Update: Thank you to the people that sent me a copy of the paper. I actually wasn't expecting people to, or else I'd have asked using something much politer than the "I wanna" locution. But it's very much appreciated - I'll try to blog on the paper soon.

Labels: Set Theory

In 'Broadening the Iterative Conception of Set', Mark Sharlow explores what he regards as an alternative version of the iterative conception which gives up the requirement of well-foundedness - the minimally iterative conception of set. The standard formulation of the iterative conception is usually though to motivate Z; interestingly, Sharlow suggests and argues that his alternative conception motivates the axioms of Quine's NF, which he notes is a system often thought to lack any intuitive motivation.

My question here is just; is the minimally iterative conception recognizable as genuinely a version of the iterative conception? Is adoption of some version of the axiom of foundation really such an add on to the conception? One hardly gets that impression from reading the literature. Michael Potter goes so far as to say outright that foundation is 'the key to the iterative conception'. (1993: 180). Charles Parsons is, if anything, even more explicit:

'One can state in approximately neutral fashion what is essential to the 'iterative' conception: sets form a well-founded hierarchy in which the elements of a set precede the set itself. In axiomatic set theory, this idea is most directly expressed by the axiom of foundation, which says that any non-empty set has an '[membership]-minimal' element.' (1977: 503-4)

Last, but never least, George Boolos wrote:

'Whatever tenuous hold on the concepts of set and member were given one by Cantor's definition of "set" and one's ordinary understanding of "element," "set," "collection," etc. is altogether lost if one is to suppose that some sets are members of themselves.' (1971: 17-8)

Boolos isn't quite explicit on this point, but it seems reasonable to take this remark as suggesting that much of the iterative conception's claim to be our intuitive pre-axiomatic conception of set stems from this requirement of well-foundedness.

So at the very least there should be an initial presumption against Sharlow's suggestion that we really are dealing with a version of the iterative conception. So what grounds does Sharlow offer in its favor?

The modification to the iterative conception Sharlow explores is a weakening of the requirement that a set is always formed at a later stage than all of its members; on the mininally iterative conception, a set constructed at some stage can turn out to be identical to one of its members formed at some earlier level. Sharlow's first point in favor of regarding the minimally iterative conception as an iterative conception is that the idea that a set formed at some stage in the hierarchy might be formed again at a later stage, far from being foreign, is built into the standard iterative conception (2001: 153). The point is perhaps obscured since, as Boolos notes, it's more convenient to talk of each set only being formed once, on its birthday (Potter's term for the earliest stage at which a set is formed):

'According to this description, sets are formed over and over again: in fact, according to it, a set is formed at every stage later than that at which it is first formed. We could continue to say this if we liked; instead we shall say that a set is formed only once, namely, at the earliest stage at which, on our old way of speaking, it would have been said to be formed.' (Boolos 1991: 19)

Secondly, Sharlow argues (2001: 157-9) that the minimally iterative conception, like the standard version, doesn't allow the Russell set to be formed. This seems to me a revealing point. Advocates of the iterative conception have often claimed that it has a very different status to other post-paradox conceptions of set, in that it might have been formulated and found attractive even had we never discovered the paradoxes; in contrast to other conceptions which are not, as Russell famously put it, 'such as even the cleverest logician would have thought of if he had not known of the contradictions' (Boolos 1971: 17). The claim put forward is that the iterative conception is our intuitive conception of set, and that in justifying particular axioms, we get grounds to take those axioms to be true - worries concerning their consistency are settled derivatively. And we've seen that many philosophers have taken an essential part of the iterative conception's claim to be a good candidate for our conception of set to be the requirement of well-foundedness.

So there's a feeling that Sharlow has understated how closely tied the iterative conception and foundation are, and that he's missed how the iterative conception is supposed to avoid paradox. It's not merely that collections like the Russell set are never formed (though of course that's a very nice feature of the iterative conception); rather, insofar as the iterative conception has a genuine claim to be our intuitive conception of set, we have good grounds to hold the axioms it motivates to be true, and hence consistent. In sum then, I'm inclined to think that some version of foundation is essential to the iterative conception, and that we need to recognize the role it plays in the project supporters of the iterative conception have embarked on. So we should be suspicious of claims that we can treat some non-well-founded sets in our set theory while still holding onto a version of the iterative conception, and still retaining the iterative conception's ability to reassure us we're not going to run into paradox.

Labels: Philosophy of Mathematics, Set Theory

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?)

Labels: Philosophy of Mathematics, Set Theory