Re: Proper classes

On Tue, 13 Dec 2005 14:03:42 -0500, Stephen J. Herschkorn wrote:
> zuhair wrote:

>>Robert Low wrote:
>>
>>
>>>zuhair wrote:
>>>
>>>
>>>>What is the cardinality of the proper class of all sets.
>>>>
>>>>
>>>The same as it was the last time you asked. It doesn't have
>>>one.
>>>
>>>
>>
>>I want to know why it does not have one.
>>
>>Why sets has cardinality , while proper classes doesn't.
>>

> Cardinality is defined in terms of functions, in particular,
> bijections. Two sets have the same cardinality iff there exists a
> bijection from one to the other. In ZFC, we define the cardinality of a
> set to be the unique cardinal number (= the least ordinal number) whence
> there exists a bijection to the set.

> Functions themselves are sets and are defined in terms of sets. (A
> function from A to B is a subset of A x B with certain
> properties.) There exist no functions from category to category, since
> that is undefined.

> I know little of category theory. From what I've seen, I guess (but am
> not at all sure) that there are things called functors which roughly
> correspond to functions, but I have no idea if there are concepts
> corresponding to injections and bijections.

A functor is essentially a function that maps a category to a category.
For example, there is a functor that maps the category of topological
spaces and homeomorphisms to the category of groups and
group-isomorphisms by associating each space with its homology group.

Although it is possible to extend the notion of a "function" so that its
"domain" and "codomain" are allowed to be proper classes, my recollection
is that the concept is not particularly interesting from the viewpoint of
cardinality, since all proper classes would turn out to have the same
"cardinality" in this extended sense.


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

Relevant Pages

  • Re: Proper classes
    ... > zuhair wrote: ... >>Why sets has cardinality, while proper classes doesn't. ... > corresponding to injections and bijections. ...
    (sci.math)
  • Re: Distinct linear orderings on Z
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  • Re: abundance of irrationals!)
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  • Proper classes (was Re: Raatikainens critique of Chaitin)
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  • Re: Distinct linear orderings on Z
    ... > Cardinality works for finite sets. ... >> It provides an analysis of what counting amounts to, ... > The one issue, however, is that infinite sets were defined by someone in ... But of course, bijections implicitly appear in that definition, too. ...
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