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
# posted by Aidan @ 12:24 PM 
Comments:
<< Home
Like last time we discussed this, my intuition is just that the self-membered thing you want to talk about just isn't a thing. Therefore, it's not available to be one of the things that form a set in either statement of the principle.
Maybe the Sharlow version of iterativeness fits your intuitions about what the consequences of these claims are?
Maybe the Sharlow version of iterativeness fits your intuitions about what the consequences of these claims are?
Sorry Kenny, I hadn't realised that was what you were driving at in your previous comment.
I think you're definitely right that there are some very tricky ontological issues looming here, and perhaps I've been guilty of over-simplifying things. You're comment about my intuitions perhaps tracking something closer to Sharlow's conception does remind me of a general worry, pushed by Maddy in 'Proper Classes' against certain views of classes. The worry, in its generalized form is if you commit to the existence of collections which aren't sets, and you believe they behave in certain set-like manners (entering into membership relations, perhaps even forming some kind of hierarchy), you'll have a hard time saying in some principled way where the line should be drawn between the sets and the non-sets.
I'm probably running the risk of facing some version of this challenge in committing myself to the existence of collections such as x, but (contra Sharlow) denying that these things should be thought of as sets. You've got me worried there's not as much distance between myself and Sharlow as I'd like. I'll need to think some more about these issues.
I think you're definitely right that there are some very tricky ontological issues looming here, and perhaps I've been guilty of over-simplifying things. You're comment about my intuitions perhaps tracking something closer to Sharlow's conception does remind me of a general worry, pushed by Maddy in 'Proper Classes' against certain views of classes. The worry, in its generalized form is if you commit to the existence of collections which aren't sets, and you believe they behave in certain set-like manners (entering into membership relations, perhaps even forming some kind of hierarchy), you'll have a hard time saying in some principled way where the line should be drawn between the sets and the non-sets.
I'm probably running the risk of facing some version of this challenge in committing myself to the existence of collections such as x, but (contra Sharlow) denying that these things should be thought of as sets. You've got me worried there's not as much distance between myself and Sharlow as I'd like. I'll need to think some more about these issues.
I've just started reading Lear's 'Sets and Semantics', and he raises the worry I focused on in my last reply to Kenny. He puts it in this wonderful way:
'One has the distinctly uneasy feeling not that we have created an iterative Frankenstein, but that we have simply lost a clear understanding of what is going on.' (87)
Post a Comment
'One has the distinctly uneasy feeling not that we have created an iterative Frankenstein, but that we have simply lost a clear understanding of what is going on.' (87)
<< Home
