Re: Distinct linear orderings on Z

From: Dave Seaman (dseaman_at_no.such.host)
Date: 03/20/05 

Date: Sun, 20 Mar 2005 22:17:35 +0000 (UTC)

On Sun, 20 Mar 2005 12:49:34 -0500, Allan C Cybulskie wrote:

> "Dave Seaman" <dseaman@no.such.host> wrote in message
> news:d1k3b3$5kb$1@mailhub227.itcs.purdue.edu...
>> On Sun, 20 Mar 2005 06:43:15 -0500, Allan C Cybulskie wrote:

>> >> I think you must have misunderstood the arguments, because the whole
>> >> point is that changing the order (or even ignoring the order
> altogether)
>> >> does *NOT* change the cardinality.

>> > Um, yeah ... but how can you claim that I misunderstood the arguments by
>> > arguing about what the properties of cardinality are when I was talking
>> > about "number of elements"? That conflates the two, but that was not
> what
>> > was going on in the thread (at least at some point).

>> > The argument actually was that counting the sets in different orders
> would
>> > come up with different answers, and was a justification for the
> conflation
>> > of cardinality with number of elements.

>> If "number of arguments" does not mean "cardinality", then what could it
>> possibly mean?

> The number of distinct entity-like things in the set, perhaps?

Can you give an example of a set that contains things that are not
distinct? Answer: no, because of the axiom of extensionality.

Can you give an example of a member of a set that is not an "entity-like
thing"? Answer: maybe, depending on whether you consider a set to be an
"entity-like thing", a term you have not defined. Note that the members
of a set are themselves sets according to ZF.

>> What is your definition? Is it a property of sets, or is
>> it a property of ordered sets?

> I think it would be properly called a property of containers, of which sets
> can be considered to be one of.

Ok, so a set is an example of a container. Is an ordered set also an
example of a container? If so, then you have simply evaded my question.
I'll ask it another way. Does changing the order of a set change the
"number of elements" in the set? If your answer is "yes", it means that
"number of elements" is a property of ordered sets. If your answer is
"no", it means that "number of elements" is a property of sets.

So what is your definition of "number of elements"? 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>