Thursday, October 11, 2007
Buffalo buffalo buffalo buffalo buffalo buffalo buffalo
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.'
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
# posted by Aidan @ 9:36 PM 
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Hi Aidan, How are you doing? hope everything fine, besides the computer...
I was reading this post of yours, and wondering what you mean when you say that that conception of sets, the limitation of interation conception, rests on assumptions which are justified if at all only on pragmatic grounds. Are you there just essentially elaborating on the complaint that you mention, that this conception is just a technical device designed to avoid paradoxes, or you have something more (or different) in mind? I would be interested in reading your paper, in this latter case.
carlotta
I was reading this post of yours, and wondering what you mean when you say that that conception of sets, the limitation of interation conception, rests on assumptions which are justified if at all only on pragmatic grounds. Are you there just essentially elaborating on the complaint that you mention, that this conception is just a technical device designed to avoid paradoxes, or you have something more (or different) in mind? I would be interested in reading your paper, in this latter case.
carlotta
# posted by 
: 11/09/2007 5:56 PM
: 11/09/2007 5:56 PM
Hey Carlotta,
What I had in mind is that some of the assumptions seem like their sole motivation is to render this hybrid conception of set stable, or to enable it to justify certain axioms. Whereas the project was originally billed as that of just clarifying and elucidating our conception of set.
I owe you a Facebook message, but it'll have to wait until I get through this assignment.
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What I had in mind is that some of the assumptions seem like their sole motivation is to render this hybrid conception of set stable, or to enable it to justify certain axioms. Whereas the project was originally billed as that of just clarifying and elucidating our conception of set.
I owe you a Facebook message, but it'll have to wait until I get through this assignment.
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