tag:blogger.com,1999:blog-4987609114415205593.post2331090158536297560..comments2024-03-28T13:40:26.497+00:00Comments on M-Phi: Uniqueness in Structural Set TheoriesJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-4987609114415205593.post-15790047322626605752016-04-18T12:12:29.704+01:002016-04-18T12:12:29.704+01:00Your structural theories tell us how to make a bui...Your structural theories tell us how to make a building structure designee with easy way thanks for share it <a href="http://www.businessmanagementpersonalstatement.com/personal-statement-for-business-management-course-by-professionals/" rel="nofollow">personal statement business management</a> .Allen jeleyhttps://www.blogger.com/profile/10312119051975318074noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-48394498353217124652013-07-22T02:08:22.701+01:002013-07-22T02:08:22.701+01:00Aldo, no, because while atoms/urelemente are empty...Aldo, no, because while atoms/urelemente are empty, they're not sets (or classes). The point of permitting atoms/urelemente is to make set theory applicable to non-sets. In fact, I'd like to treat sui generis mathematicalia as atoms, so every sui generis abstractum $a$ (e.g., a pair, or a cardinal, or a real) is "empty" in the sense that, for all $x$, $x \notin a$.<br /><br />But, at the moment, I'm trying gradually to understand the basic picture here behind the notions of a "structural set" versus a "material set", without delving into the complicated algebra used to model subsets, power sets, etc. But I do have an intuition that $\varnothing$ should be unique simpliciter, rather than unique up to isomorphism.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-72857780976051650422013-07-22T01:36:04.269+01:002013-07-22T01:36:04.269+01:00Jeffrey, as to whether structural set theory allow...Jeffrey, as to whether structural set theory allows multiple empty sets: do you have the same reservations about ZFU, where the different Urelements are -- formally -- just distinct memberless items?Aldo Antonellihttp://aldo-antonelli.orgnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-73553272774198873122013-07-21T22:53:01.096+01:002013-07-21T22:53:01.096+01:00Antonio,
Right - I think that is indeed the answe...Antonio,<br /><br />Right - I think that is indeed the answer, and reading around other formulations (e.g., Makkai's), the non-uniqueness is mandated in structural set theories.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-90654438326334069572013-07-21T22:12:57.518+01:002013-07-21T22:12:57.518+01:00Hi Jeffrey, Thanks.
Yes, the quote was indeed sup...Hi Jeffrey, Thanks. <br />Yes, the quote was indeed supposed to answer not the issue of uniqueness, but maybe only your doubt "whether forbidding formulas of the form A=B is mandated by the motivating philosophical/conceptual thought behind structural set theory, or whether it is largely a matter of convenience.". And to me that passage seems to allow for the first answer. <br /><br />As for uniqueness, at least for ETCS, here they say that "by ∅ we mean an initial object. I mean any initial object (there can be lots!), although any two are uniquely isomorphic." (from http://boolesrings.org/asafk/2013/on-leinsters-rethinking-set-theory/)<br /><br />AntonioAntonio Negrohttps://www.blogger.com/profile/08533947260637745377noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-49716719880167292012013-07-21T18:09:10.727+01:002013-07-21T18:09:10.727+01:00Thanks, Antonio
Yes, I know that passage; it's...Thanks, Antonio<br />Yes, I know that passage; it's very similar to the one I quote above too, from Mike Shulman: <br /><br />"no thing can be given to you as an element of more than one structural-set."<br /><br />But it doesn't really seem to answer the question concerning uniqueness of $\varnothing$. <br /><br />Suppose $\varnothing_1 : \mathsf{Set}$ and $\varnothing_2: \mathsf{Set}$ satisfy the conditions of being *an* empty set. These structural sets tabulate empty relations $\varphi : A \looparrowright B$, where $\varphi(x,y)$ never holds for $x:A$, $y:B$. These empty sets $\varnothing_1$ and $\varnothing_2$ will be isomorphic or "equivalent", in some relevant category-theoretic sense. <br /><br />But I want to know if we have:<br /><br />$\varnothing_1 = \varnothing_2$?<br /><br />Are they literally the same thing? Or merely "isomorphic"?<br /><br />Cheers,<br /><br />Jeff<br />Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-35226049550244381132013-07-21T14:09:25.860+01:002013-07-21T14:09:25.860+01:00"What's unclear to me then, at bottom, is..."What's unclear to me then, at bottom, is whether forbidding formulas of the form A=B is mandated by the motivating philosophical/conceptual thought behind structural set theory, or whether it is largely a matter of convenience." <br /><br />Jeffrey, Maybe this answers your doubt? <br />(from http://ncatlab.org/nlab/show/structural+set+theory):<br /><br />"Structural set theory provides a foundation for mathematics which is free of this “superfluous baggage” attendant on theories such as ZF, in which there is lots of information such as whether or not 3∈17 (yes, says von Neumann; no, says Zermelo) which is never used in mathematics. In a structural set theory, the elements (such as 3) of a set (such as ℕ) have no identity apart from their existence as elements of that set, and whatever structure is given to that set by the functions and relations placed upon it. That is, sets (together with other attendant concepts such as elements, functions, and relations) are the “raw material” from which mathematical structures are built. By contrast, theories such as ZF may be called material set theories or membership-based set theories.<br /><br />Thus, somewhat paradoxically, it turns out that one of the primary attributes of a structural set theory is that the elements of a set have no “internal” structure; they are only given structure by means of functions and relations. In particular, they are not themselves sets, and by default cannot be elements of any other set (not in the sense that it is false that they are, but in the sense that it is meaningless to ask whether they are), ***so that elements of different sets cannot be compared*** (unless and until extra structure is imposed). Structural set theory thus looks very much like type theory. We contrast it with material set theories such as ZF, in which the elements of sets can have internal structure, and are often (perhaps always) themselves sets."<br />Antonio Negrohttps://www.blogger.com/profile/08533947260637745377noreply@blogger.com