Monday, February 19, 2007
What is essential to the iterative conception of set?
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.
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
# posted by Aidan @ 5:33 PM 
Comments:
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Any standard notion of the iterative conception clearly must use foundation. I wasn't sure what it could possibly mean to have something like an iterative conception without foundation, so I read the paper.
It looks to me that he's focusing on a feature of the iterative conception that I would have rather called "combinatorial" or something rather than "iterative". "Iterative" definitely suggests doing one thing and then another, whereas "combinatorial" expresses the notion of arbitrary combination, which seems closer to what he has in mind. His point just seems to be that you can gather some things together into a set once you're sure they're going to exist, even if they don't already exist (where this temporal vocabulary may or may not really be taken seriously).
However, I wasn't terribly convinced that there's really a principled distinction here with these "non-self-defeating properties". It almost looks like he was coming down to saying that some things could be gathered into a set iff there is an algorithm for picking them out, given a way to pick out the set that is currently being created. But I don't think he'd want to put it that way.
Also, some other points. He suggests that the set-theoretic paradoxes all involve some form of self-reference - I wonder if something like Yablo's paradox could be created here as well? Also, in his restriction on comprehension principles, I wonder if something like the series of bad company objections could be raised against his restrictions. (That is, two properties, neither self-defeating, such that if both pick out a set then there is a contradiction.)
And finally, his justification for NF is very roundabout. Compared to all the talk of how to ban properties identifying a set with a non-member, the stratification restriction looks like a much more natural way to avoid paradox.
Which just raises the question as to why this is any more intuitive a justification for NF than the original idea.
It looks to me that he's focusing on a feature of the iterative conception that I would have rather called "combinatorial" or something rather than "iterative". "Iterative" definitely suggests doing one thing and then another, whereas "combinatorial" expresses the notion of arbitrary combination, which seems closer to what he has in mind. His point just seems to be that you can gather some things together into a set once you're sure they're going to exist, even if they don't already exist (where this temporal vocabulary may or may not really be taken seriously).
However, I wasn't terribly convinced that there's really a principled distinction here with these "non-self-defeating properties". It almost looks like he was coming down to saying that some things could be gathered into a set iff there is an algorithm for picking them out, given a way to pick out the set that is currently being created. But I don't think he'd want to put it that way.
Also, some other points. He suggests that the set-theoretic paradoxes all involve some form of self-reference - I wonder if something like Yablo's paradox could be created here as well? Also, in his restriction on comprehension principles, I wonder if something like the series of bad company objections could be raised against his restrictions. (That is, two properties, neither self-defeating, such that if both pick out a set then there is a contradiction.)
And finally, his justification for NF is very roundabout. Compared to all the talk of how to ban properties identifying a set with a non-member, the stratification restriction looks like a much more natural way to avoid paradox.
Which just raises the question as to why this is any more intuitive a justification for NF than the original idea.
Kenny,
I haven't read all of it, but you might be interested in Keith Simmons' 'A Berry and a Russell Without Self-Reference', in Phil Studies 2005, for a development of your Yablo-instinct.
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I haven't read all of it, but you might be interested in Keith Simmons' 'A Berry and a Russell Without Self-Reference', in Phil Studies 2005, for a development of your Yablo-instinct.
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