Editing Class (set theory) (section)

== Paradoxes ==
The [[naive set theory#Paradoxes|paradoxes of naive set theory]] can be explained in terms of the inconsistent [[tacit assumption]] that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest [[proof (mathematics)|proof]]s that certain classes are proper (i.e., that they are not sets). For example, [[Russell's paradox]] suggests a proof that the class of all sets which do not contain themselves is proper, and the [[Burali-Forti paradox]] suggests that the class of all [[ordinal numbers]] is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes.  A [[conglomerate (category theory)|conglomerate]], on the other hand, can have proper classes as members.<ref>{{citation |last1=Herrlich |first1=Horst |author-link1=Horst Herrlich|last2=Strecker |first2=George |year=2007 |title=Category theory |chapter=Sets, classes, and conglomerates |chapter-url=http://www.heldermann.de/SSPM/SSPM01/Chapter-2.pdf |url=http://www.heldermann.de/SSPM/SSPM01/sspm01.htm |publisher= Heldermann Verlag |edition=3rd |pages=9–12}}</ref>