Tuesday, February 27, 2007
Is Limitation of Size a consequence of the Iterative Conception?
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?
'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
If it's in Vogue..........
No sets please, we're British
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.
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
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
Wednesday, February 14, 2007
Overly strong statement of the day 2
Well, this isn't really a sequel, since the original installment was Shawn's not mine. But anyways, I found this pretty hard to stomach:
'There is no market without government and no government without taxes; and what type of market there is depends on laws and policy decisions that government must make. In the absence of a legal system supported by taxes, there couldn’t be money, banks, corporations, stock exchanges, patents, or a modern market economy – none of the institutions that make possible the existence of almost all contemporary forms of income and wealth.
It is therefore logically impossible that people should have any kind of entitlement to all of their pretax income.'
(Thomas Nagel and Liam Murphy, The Myth of Ownership: 32. My italics.)
Call me old-fashioned.....
'There is no market without government and no government without taxes; and what type of market there is depends on laws and policy decisions that government must make. In the absence of a legal system supported by taxes, there couldn’t be money, banks, corporations, stock exchanges, patents, or a modern market economy – none of the institutions that make possible the existence of almost all contemporary forms of income and wealth.
It is therefore logically impossible that people should have any kind of entitlement to all of their pretax income.'
(Thomas Nagel and Liam Murphy, The Myth of Ownership: 32. My italics.)
Call me old-fashioned.....
Labels: Overly Strong Statements
Sunday, February 11, 2007
Self-Reference Without Paradox
My friend Nolan just sent me a link to these pictures of a self-referential warning sign. The intrigued/bored should check them out.
Saturday, February 10, 2007
Super Soaker
I've been a little neglectful here the past couple of weeks, due to work for classes and TAing, refereeing papers for the grad conference, and, well, life. I'll be reading a lot on the epistemology of language this weekend, so I'll hopefully have some things to post about after that. In the meantime, here's xkcd's recent comic 'Philosophy':
Labels: xkcd
Friday, February 02, 2007
Arche PhD Studentships
UNIVERSITY OF ST ANDREWS
SCHOOL OF PHILOSOPHICAL, ANTHROPOLOGICAL AND FILM STUDIES
Arché: AHRC Centre for the Philosophy of Logic, Language, Mathematics and Mind
PhD Studentships
Arché is offering up to six three-year PhD studentships for uptake from September 2007. The studentships are primarily intended to support doctoral research in the broad areas of contemporary epistemology and the philosophy of language, although applications will be considered from well-qualified candidates with research interests in any area within the Centre’s broad remit. Successful applicants can expect to work under the supervision of the Arché Professors Jessica Brown, Herman Cappelen and Crispin Wright.
Two of the studentships are open to applicants of any nationality; they provide a yearly maintenance grant of up to £12 500 and full coverage of tuition fees. A further two studentships are likewise open to all applicants and provide full fees but a maintenance award at a lower level. The remaining two studentships are associated with the AHRC funded Basic Knowledge research project. These studentships, which are open only to UK/EU applicants, provide full coverage of tuition fees and a maintenance grant of up to £12,500 per year for a maximum of three years (subject to satisfactory progress).
Closing date for receipt of applications: 7th March 2007
For the full advertisement and instructions on how to apply see:
http://www.st-andrews.ac.uk/~arche
SCHOOL OF PHILOSOPHICAL, ANTHROPOLOGICAL AND FILM STUDIES
Arché: AHRC Centre for the Philosophy of Logic, Language, Mathematics and Mind
PhD Studentships
Arché is offering up to six three-year PhD studentships for uptake from September 2007. The studentships are primarily intended to support doctoral research in the broad areas of contemporary epistemology and the philosophy of language, although applications will be considered from well-qualified candidates with research interests in any area within the Centre’s broad remit. Successful applicants can expect to work under the supervision of the Arché Professors Jessica Brown, Herman Cappelen and Crispin Wright.
Two of the studentships are open to applicants of any nationality; they provide a yearly maintenance grant of up to £12 500 and full coverage of tuition fees. A further two studentships are likewise open to all applicants and provide full fees but a maintenance award at a lower level. The remaining two studentships are associated with the AHRC funded Basic Knowledge research project. These studentships, which are open only to UK/EU applicants, provide full coverage of tuition fees and a maintenance grant of up to £12,500 per year for a maximum of three years (subject to satisfactory progress).
Closing date for receipt of applications: 7th March 2007
For the full advertisement and instructions on how to apply see:
http://www.st-andrews.ac.uk/~arche
Labels: Misc
