Re: Distinct linear orderings on Z

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

Date: Mon, 21 Mar 2005 10:50:21 -0500

Dave Seaman said:
> On Sun, 20 Mar 2005 12:02:29 -0600, Albert Wagner wrote:
> > Dave Seaman wrote:
> >> On Sun, 20 Mar 2005 10:30:49 -0600, Albert Wagner wrote:
> >>
> >>>Basically, Dave, we simply don't like your definitions. And it
> >>>being a fact that you have no legal protection nor moral high
> >>>ground for your definitions we are free to make up any we like
> >>>better. The consensus of those who erroneous believe that they
> >>>have a patent on numbers and their use carries no weight with
> >>>everyone else. Mathematics is a commons, public property. e.g.
> >>>cardinality has to do with redness, nothing more.
> >>
> >>
> >> You have every right to make up your own definitions, but I notice that you
> >> have ducked the question yet again. I asked what your definition is. No
> >> reply.
>
> > That's because it wasn't me that you asked.
>
> You said "we don't like your definitions", so I thought you were speaking
> for the group.
>
> Never mind, I am asking you now. For you and for anyone else who claims
> "number of elements" does not mean "cardinality": What is your
> definition of "number of elements"?
>
>
>
>
I think the normal definition is good. Count the elements. When the set becomes
ifinite that's impossible, so cardinality was invented to virutally enumerate
the elements using mapping functions between sets of numerical elements. When
using numerical elements, such as integers, rationals or reals, each number
type saturates the real number line to a different extent. Personally, I see
the rationals as being more similar to the reals that integers in this respect,
since they fully saturate the line from a finite perspective (though perhaps
porously), which integers do not. In any case, when you are looking at infinite
sets of numbers in finite sections of the number line that have the same
saturation on the number line, such as [0,1] and [2,4] in the reals, the
relative sizes of the sets should be considered to be the same as their
relative covered sections of the number line. I'd say the size of the set
should be the product of its saturation and domain on the number line. The
domain may be easy to measure. The saturation is a numerical representation of
something like order type, and needs to be formally defined. 

-- 
Smiles,
Tony