Re: Distinct linear orderings on Z

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/19/05 

Date: Sat, 19 Mar 2005 07:30:49 -0500

"Stephen J. Herschkorn" <sjherschko@netscape.net> wrote in message
news:423BE030.9020308@netscape.net...
> Tony Orlow (aeo6) wrote:
>
> Your flaw is assuming that you can talk of the cardinality of infinite
> sets in the same way as you can speak of finite sets.

Um, that seems to be the problem we are having with those who maintain
cardinality as being the number of elements on infinite sets, since there
seems little reason to assume that the mapping approach will work on
infinite sets as well as on finite sets.

  Do you think
> that N = {0, 1, 2, 3,...} has a larger cardinality that N \ {0} = {1,
> 2, 3,...}? Or that listing the set {1, 2, 3} as {2, 3, 1} changes it
> cardinality? Does listing N as {0, 1, 2, 4, 3, 6, 8, 10, 5, 12, 14,
> 16, 18, 7,...} change its cardinality? Do you really think that the
> cardinality of [0,2] is larger than the cardinality of [0,1]?

The difficulty here is that neither Tony nor I really think that about
CARDINALITY, but think that about NUMBER OF ELEMENTS. I am willing to
accept that the cardinality is as you say in all of the above cases, but
believe that the number of elements changes for the first and last examples
(I don't think that ordering changes the number of elements in a set). And
the only time Tony really talked about ordering is when other people argued
that they could change the ordering of a set and thus change the number of
elements, which seemed to make little sense.

Relevant Pages

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