From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/205 Path: news.gmane.org!not-for-mail From: Meredith Gregory Newsgroups: gmane.science.mathematics.categories Subject: Re: naive questions about sets Date: Fri, 27 Mar 2009 18:07:12 -0700 Message-ID: Reply-To: Meredith Gregory NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1238247120 30818 80.91.229.12 (28 Mar 2009 13:32:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 28 Mar 2009 13:32:00 +0000 (UTC) To: Toby Bartels , categories@mta.ca Original-X-From: categories@mta.ca Sat Mar 28 14:33:18 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LnYey-0001E4-SL for gsmc-categories@m.gmane.org; Sat, 28 Mar 2009 14:33:17 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LnXw4-000264-9f for categories-list@mta.ca; Sat, 28 Mar 2009 09:46:52 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:205 Archived-At: Toby, Thanks for your note. A longer reply is forthcoming, but in regards to notation, in CS-y conventions it is now standard to separate out the "free" syntax from the relations. So, you're right, of course, that the * denotes a list. (It originates from the Kleene operator.) The standard treatment is to whack this down by what is called a structural equivalence relation. In this case, you have something like n1,n2 = n2,n1 n1,n1 = n1 for the structural relations. This is roughly equivalent to specifying models as algebras, i.e. a free monad plus a a structure map. i find the ghettoization of the CS notations -- which are extremely compact and well aligned with category theoretic sensibilities -- to be a source of unending frustration in communications with the larger mathematical communities. i really need to write a tutorial. Best wishes, --greg On Fri, Mar 27, 2009 at 5:35 PM, Toby Bartels < toby+categories@ugcs.caltech.edu >wrote: > Meredith Gregory wrote in part: > > >My > >daughter and i decided that what went into the physical containers were > >little promisory notes that could be redeemed for actual things. In this > >version of the construction you only get flat structures, a set never > >contains a set. That left the problem of the language in which the > promisory > >notes were written. Why not use the language of containers? > > - Container ::= {c| Note* |c} // BlackSet ::= {b| RedSet* > |b} > > - Note ::= {n| Container* |n} // RedSet ::= {r| BlackSet* > |r} > > Not to denigrate your interesting variation, > but you get a result more like traditional set theory > if you say that each container contains a *single* note > which itself has a list of containers written on it: > - Container ::= {c| Note |c} > - Note ::= {n| Container* |n} > (You could also have a list of notes, each of which has one container.) > > Of course, this is functionally equivalent to > - Container ::= {c| Container* |c} > This is the usual model of what pure sets are like, > but (as you and your daughter noted) this give an unphysical metaphor. > However, you could just as easily do things like this: > - Note ::= {n| Note* |n} > Since {n|...|n} contains things by writing down names for them, > rather than having them physically present as {c|...|c} implies, > there is no physical impossibility here. > > Incidentally, the notation X* suggests to me a list, > where order and repetition matter and only finitely many terms can appear. > Of course, sets are not like this, as you know. > So instead of X* I would write P(X), using "P" for "power". > > This matches how category theorists model pure sets > (the sets that appear in ZF-style set theory). > A _universe of pure sets_ consists of a set U > and a function from U to the power set P(U) of U, > or if you prefer a binary relation E (for 'epsilon') on U, > satisfying a few axioms (extensionality and well-foundedness, > although it's also interesting to consider ill-founded sets). > You get paradoxes like Russell's if you add the axiom > that the function U -> P(U) is invertible. > > Of course, I said "set" above, but there I just meant > an object of the category of sets as category theorists think of it. > That is, simply a collection of atoms that may be equal > but have no other structure (and in particular are not themselves sets). > So this shows how to model ZF-style set theory in categorial set theory. > (I apologise for repetition if you already know all about that.) > > > --Toby > -- L.G. Meredith Managing Partner Biosimilarity LLC 806 55th St NE Seattle, WA 98105 +1 206.650.3740 http://biosimilarity.blogspot.com