tag:blogger.com,1999:blog-4987609114415205593.post3770069804299356744..comments2022-01-19T03:06:39.163+00:00Comments on M-Phi: SGM: Sui Generis MathematicsJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-4987609114415205593.post-62248539908953696842013-07-15T19:55:49.821+01:002013-07-15T19:55:49.821+01:00Thanks, Dennis
Glad to hear you're interested...Thanks, Dennis<br /><br />Glad to hear you're interested! Yes, anti-reductionism is part of the motivation. But I think there's no additional need to eliminate sets. The foundational/base part is really the system $\mathsf{ASC}$ ("atoms, sets and classes"), here,<br /><br />http://m-phi.blogspot.co.uk/2013/07/asc-atoms-sets-and-classes.html<br /><br />But this theory $\mathsf{ASC}$ (of classes) is very weak, and doesn't imply the existence of any sets. To get the existence of sets, one has to add specific set-existence axioms saying things like "the class $\{x \mid x \neq x\}$ is a set", etc.<br /><br />Everything else---i.e., the sui generis abstracta---is then added by abstraction principles. The difference with recent neo-logicism is that one has given up on the idea of generating sets from abstraction principles. As sui generis abstracta, they needn't be reduced to sets; but they can be (to check consistency, say); and the reduction to some set is like a "gauge choice".<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-77683397982685556182013-07-15T19:44:24.016+01:002013-07-15T19:44:24.016+01:00Thanks for the reply Jeff.
Does the usage of set ...Thanks for the reply Jeff.<br /><br />Does the usage of set theory as the base theory to be extended commit you to some view of set theory as "foundational"? Clearly not in the sense of being a reduction base, but it seems to me there still might be some priority claim you're committed to in virtue of utilizing set theory as the base theory. This is only a hazy worry though, and perhaps something that can be easily skirted.<br /><br />I am actually rather sympathetic to this sort of anti-reductionist project. I look forward to future posts (and/or publications) on this topic.Dennis Kavlakoglunoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-9360033657491063892013-07-15T19:20:50.858+01:002013-07-15T19:20:50.858+01:00Thanks, Dennis
Actually, the theory itself is alr...Thanks, Dennis<br /><br />Actually, the theory itself is already sort-of-second order, because it starts with a theory of classes & atoms, with impredicative comprehension (I call this ASC). Sets are defined to be classes that are members. <br /><br />Usually, abstractionists/neo-logicists have tried to get sets from SOL using a refined abstraction principle (a modification of Frege's BLV); but they haven't been successful, because to get something like the cumulative hierarchy of sets out of SOL + abstraction for sets, one has to make complicated structural assumptions built into the abstraction principle, which are more or less equivalent to Zermelo's axioms.<br /><br />So, instead, I don't try and reduce sets to extensions of classes, but leave them as they are. So, I can treat sets themselves as sui generis mathematical objects (given by set existence axioms, a la Zermelo); and then all the other abstracta arise from appropriate abstraction principles.<br /><br />The point isn't to eliminate sets; the point is to extend the set-theoretical universe with sui generis abstracta, so that arbitrary reductions are not required (except to check consistency/conservation).<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-5414410060412351492013-07-15T19:00:26.638+01:002013-07-15T19:00:26.638+01:00"Extend set theory with abstraction principle..."Extend set theory with abstraction principles for sui generis mathematical objects."<br /><br />But doesn't this give rise to another charge of arbitrariness? Why choose set theory as the base theory to be extended? Why not embed the abstraction principles in a second-order logic (a la Neo-Fregeanism) this way you can at least avoid the charge that you're privileging one mathematical theory over another? Dennis Kavlakoglunoreply@blogger.com