User bernhard stadler - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:37:49Z http://mathoverflow.net/feeds/user/23693 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63265/what-are-maps-between-proper-classes/98472#98472 Answer by Bernhard Stadler for What are "maps" between proper classes? Bernhard Stadler 2012-05-31T10:23:01Z 2012-05-31T10:23:01Z <p>Instead of MK set theory + Morse ordered pair definition, is ARC set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, <a href="http://en.scientificcommons.org/49425946" rel="nofollow">Bibliography</a> <a href="http://www.phys.uu.nl/~wwwgrnsl/muller/SetClassCat-BJPS2001.pdf" rel="nofollow">PDF</a>) with the usual Kuratowski ordered pair an option? </p> <p>ARC supposedly proves the existence of the $n$-th power-class of the set universe V for any <code>$n \in \mathbb{N}$</code>, and all so-called "good" classes provably exist, good classes being the class of all sets and "the powerclass and the union-class of a good class, and the union-class, the intersectionclass, the complement-class, the pair-class, the ordered pair-class, and the Cartesian product-class of any finite number of members of one good class".</p> <p>According to the cited paper, ARC is consistent relative to ZFC plus a strongly inaccessible cardinal axiom. I think that this set theory looks quite nice, so I'm wondering why it didn't take off at all. The formal proofs should be in Muller's PhD thesis, which I don't have access to.</p> http://mathoverflow.net/questions/67786/bijection-of-proper-classes/98350#98350 Answer by Bernhard Stadler for Bijection of proper classes Bernhard Stadler 2012-05-30T09:59:36Z 2012-05-30T09:59:36Z <p>ARC, an extension of Ackermann set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, <a href="http://en.scientificcommons.org/49425946" rel="nofollow">http://en.scientificcommons.org/49425946</a> <a href="http://www.phys.uu.nl/~wwwgrnsl/muller/SetClassCat-BJPS2001.pdf" rel="nofollow">PDF</a>) proves the existence of the n-th powerclass of the set universe V for any <code>$n \in \mathbb{N}$</code>. That should make it possible to define pairs/triples of classes and thus functions and bijections the usual way.</p>