From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7455 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: An internal definition in a realizability topos Date: Mon, 24 Sep 2012 11:52:35 -0400 Message-ID: Reply-To: Andrej Bauer NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1348510478 25020 80.91.229.3 (24 Sep 2012 18:14:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 24 Sep 2012 18:14:38 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Mon Sep 24 20:14:43 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.134]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1TGDB3-0007GZ-Rj for gsmc-categories@m.gmane.org; Mon, 24 Sep 2012 20:14:41 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51950) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1TGDAf-0000s3-El; Mon, 24 Sep 2012 15:14:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TGDAf-0007KR-Gt for categories-list@mlist.mta.ca; Mon, 24 Sep 2012 15:14:17 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7455 Archived-At: Consider a realizability topos RT(A) over some PCA A. There is an embedding Nabla : Set -> RT(A), which takes a set X to the assembly whose underlying set is X and the existence predicate is trivial, i.e., every element of X is realized by every realizer of A. Let [n] = {0, 1, 2, .., n-1} be the set with n elements. I have some interest in the subobject of Nabla([n]) whose underlying set is [n] and the existence predicate is E(0) = { numeral(0) } E(1) = { numeral(0), numeral(1) } ... E(k) = { numeral(k-1), numeral(k) } ... E(n-1) = {numeral(n-2)} In words: there are n elements 0, 1, 2, ..., n-1, where two consequtive elements share a realizer. The question is; is this object definable in the internal language of the topos? Note that Nabla(2) is definable as the object of not-not-stable truth values, and that each Nabla(X) is also definable with a bit more work. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]