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

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: Using classes instead of sets
Posted: Mar 28, 2013 3:01 PM
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On 2013-03-28, <> 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.

Other than with types, which are usually avoided because of
difficulties, proper classes are not allowed to be elements.
ZF has no proper classes; the essentially equivalent NBG avoids
the known paradexes by not allowing proper classes to be elements.
With the strongest forms of the Axiom of Choice, proper classes are
all of the same size as the universe.

> Paul Epstein

This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558

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