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

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fom

Posts: 1,968
Registered: 12/4/12
Re: Using classes instead of sets
Posted: Mar 30, 2013 12:26 AM
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On 3/29/2013 5:24 PM, Shmuel (Seymour J.) Metz wrote:
> In <f8ral8lndkanp4gd70pu1fmblq4g912o74@4ax.com>, on 03/29/2013
> at 06:07 AM, quasi <quasi@null.set> said:
>

>> There are certain concepts for which sets are inadequate and classes
>> come to the rescue. For example, we need the class concept if we
>> want to define an equivalence on the collection of all groups, since
>> that collection is not a set.

>
> There's no such set in ZFC, but there are set theories in which it
> exists aqnd is a set.
>


Can you name one? I am curious.

I had begun dusting off my category theory books
to address that aspect of the question. MacLane
defines metacategories and then instantiates
categories relative to set representations. The
result is a distinction between large categories
and small categories.

Then there is the use of Grothendieck universes.
Those are essentially inaccessible cardinals as
described by a particular form introduced by
Tarski. But, even with these, there is an implied
hierarchy of universes for which no mathematics
has been known to need more than a sequence of
3 (per the account that I read).

I suppose in that regard, there may be

(some standard set theory)+(some large cardinal axiom)

combination that constitutes a set theory
different from ZFC for which your statement
holds. Is this what you had in mind?

(No need to specify if that is the case)








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