Re: Distinct linear orderings on Z

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/21/05 

Date: Mon, 21 Mar 2005 11:54:55 -0500

Dave Seaman said:
> On Mon, 21 Mar 2005 10:08:57 -0500, Tony Orlow wrote:
> > Dave Seaman said:
> >> On Sat, 19 Mar 2005 07:30:49 -0500, Allan C Cybulskie wrote:
> >>
> >> > 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.
> >>
> >> 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.
> >>
> >> Sets qua sets do not have an order of any kind. It's only when we impose
> >> some additional structure on a set that we get an order, but this is
> >> completely irrelevant to the existence of bijections and therefore the
> >> cardinality.
> >>
> >>
> >>
> >>
> > Ah, but it is you who misunderstand the argument, for the ordering does make a
> > difference which cardinality ignores, as you have just said. This is what I
> > believe causes cardinality to be interpreted to have "counterintuitive"
> > conclusions that beg correction.
>
> When I wrote that I did not understand that you were using the phrase
> "number of elements" to mean something different from cardinality. Since
> then I have asked repeatedly for a definition of "number of elements",
> but none has been forthcoming.
>
> There is a different kind of mapping called an "order isomorphism" that
> applies to ordered sets. Such a mapping is a bijection with the
> additional property that it preserves order. Therefore, you have a
> choice: you can talk about plain bijections, which determine
> cardinality, or you can talk about order-preserving bijections, which
> determine order type.
>
> By convention, mathematicians understand the phrase "number of elements"
> to refer to the former, not the latter. If you want a different meaning,
> it's best to use the technical term and talk about "order types", not the
> misleading term "number of elements".
>
>
>
Thanks for the understanding and explanation. The "order-isomorphism" you refer
to sounds much like what I am suggesting. I quickly googled the term and got a
basic definition, but I wonder if this approach has been applied to deriving
finite ratios between the sizes of infinite sets, such as between [0,1] and
[0,2], or finite offsets, such as between (0,1,2...) and (1,2,3,...)? From what
I read about order types, this extension has not been realized. Is there
another term I should be looking for?

It seems to me that the term "number of elements", being constructed from the
fairly simple concepts of "number" and "thing", is a rather general term. The
more exact term as it stands is "cardinality", so it seems to me that if that
is what one is referring to, then that's the word they should use. To equate
cardinality with "number of elements" for infinite sets, to the exclusion of
any other definitions or approaches to defining that vague term, is a mistake
as I see it. It's not that cardinality is a mistake, just that assumed
implication, and I am not at all convinced that it's a semantic or
philosophical matter, when talking about consistent mathematical results. 

-- 
Smiles,
Tony

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