From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/204 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: naive questions about sets Date: Fri, 27 Mar 2009 17:35:59 -0700 Message-ID: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1238247110 30790 80.91.229.12 (28 Mar 2009 13:31:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 28 Mar 2009 13:31:50 +0000 (UTC) To: Meredith Gregory , categories@mta.ca Original-X-From: categories@mta.ca Sat Mar 28 14:33:07 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 1LnYel-0001Ag-Nb for gsmc-categories@m.gmane.org; Sat, 28 Mar 2009 14:33:03 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LnXuk-00023d-8X for categories-list@mta.ca; Sat, 28 Mar 2009 09:45:30 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:204 Archived-At: 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