Answer by Bernhard Stadler for What are "maps" between proper classes?

2012-05-31T10:23:01Z

Instead of MK set theory + Morse ordered pair definition, is ARC set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, Bibliography PDF) with the usual Kuratowski ordered pair an option?

ARC supposedly proves the existence of the $n$-th power-class of the set universe V for any $n \in \mathbb{N}$ , 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".

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.

Answer by Bernhard Stadler for Bijection of proper classes

2012-05-30T09:59:36Z

ARC, an extension of Ackermann set theory (F.A.Muller, "Sets, Classes, and Categories", 2001, http://en.scientificcommons.org/49425946 PDF) proves the existence of the n-th powerclass of the set universe V for any $n \in \mathbb{N}$ . That should make it possible to define pairs/triples of classes and thus functions and bijections the usual way.

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Answer by Bernhard Stadler for What are "maps" between proper classes?

Answer by Bernhard Stadler for Bijection of proper classes