Kaba
Posts:
289
Registered:
5/23/11


Re: Using classes instead of sets
Posted:
Mar 28, 2013 1:21 PM


28.3.2013 18:24, pepstein5@gmail.com wrote: > On Thursday, March 28, 2013 3:11:31 PM UTC, Frederick Williams wrote: > >> ... >> If groups could have classes for the collection of their elements, and >> >> if we call such groups "Groups", then we couldn't call the collection of >> >> Groups a set or a class, could we? > ... > > I don't see why not. Without further restrictions, the collection of Groups would seem to be too big to be a set, but your Groups could form a class, I would think. Classes are allowed to contain other classes after all. Of course, we get Russelltype paradoxes if we allow entities to contain themselves, whether the entities be sets or classes.
In general you have level0 collections, which are the ordinary sets, level1 collections (classes), which are collections of level0 collections, and so on leveln collections are collections of level(n1) collections. There is an infinity of different kinds of collections. No paradoxes in that:) Classes represent a single step in this generalization.
Such ideas are needed and used in higher category theory:
http://en.wikipedia.org/wiki/Ncategory
Category theory contains ideas which generalize ordinary structures on sets to structures on categories: monoids, groups etc. So maybe have a look at category theory?
Disclaimer: I'm learning category theory at the moment, so treat my words with caution.
 http://kaba.hilvi.org

