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Topic: Using classes instead of sets
Replies: 26   Last Post: Apr 1, 2013 8:04 PM

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Posts: 289
Registered: 5/23/11
Re: Using classes instead of sets
Posted: Mar 28, 2013 1:21 PM
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28.3.2013 18:24, 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 Russell-type paradoxes if we allow entities to contain themselves, whether the entities be sets or classes.

In general you have level-0 collections, which are the ordinary sets,
level-1 collections (classes), which are collections of level-0
collections, and so on level-n collections are collections of
level-(n-1) 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:

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.


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