Re: Distinct linear orderings on Z

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/23/05 

Date: Wed, 23 Mar 2005 17:25:57 -0500

"Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
news:87psxqhnsh.fsf@phiwumbda.org...
> "Allan C Cybulskie" <allan.c.cybulskie@yahoo.ca> writes:
>
> > I'm only going to say this once:
> >
> > Here is one of my main problems with the whole cardinality as number of
> > elements thing. It seems to me that the history of this part of set
theory
> > likely ran something like this. First, we decided we wanted to know how
> > many things we had in some container, and then invented counting. We
also
> > soon after invented some other methods for determining this (weighing
and
> > dividing by weight for objects with the same weight, for example). We
also
> > understood that if someone says "if all of the things that are in one
set
> > are in another set (container) and there are more besides, then the
second
> > set has more things in it." Soon after, someone really noticed this
whole
> > number system thing and started building sets and set theory based on
that.
> > Then we noticed that functionally counting was the same thing as mapping
> > onto the set of integers. And then at some point this led us to
deciding
> > that cardinality and bijection works really well as a notion of relative
> > number of elements for finite sets. And then we started looking at what
it
> > meant for an infinite set to have a certain number of elements. And
> > mathematicians decided to apply cardinality as just being that as well
...
> > even though it contradicted the notion that obviously if one set
contains
> > everything in another set and more, it must have more elements.
>
> Sorry, but made-up history of mathematics is really James S. Harris's
> forte. Why add competition?

So what's wrong with it? At least one other mathematician seems to think it
was close.

Relevant Pages

  • Re: Distinct linear orderings on Z
    ... Since the notion of "cardinality" is not very differentiating in this ... There's no point in arguing whether cardinality or natural density ... common) that mathematicians were getting persnickety about definitions ... We think, for example, that the axioms of Zermelo and Frankl capture the ...
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  • Re: Distinct linear orderings on Z
    ... I am not insisting that mathematicians cannot use cardinality to ... >> decent equivalent to counting. ... >> capture what we feel is IMPORTANT about our counting. ...
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    ... I am not insisting that mathematicians cannot use cardinality to ... basic misunderstanding of infinite sets that it is hard to explain ... > capture what we feel is IMPORTANT about our counting. ...
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    ... :> In sci.math Tony Orlow wrote: ... :>:> cardinality? ... Tony and others were claiming that the mathematicians ... didn't sound like it applied to infinite sets anyway. ...
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