Re: Platonism
From: Stuart M Newberger (smnewberger_at_comcast.net)
Date: 12/04/04
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Date: 4 Dec 2004 15:53:15 -0800
lesterDELzick@worldnet.att.net (Lester Zick) wrote in message news:<41b28a0c.14919770@netnews.att.net>...
> On 3 Dec 2004 16:24:59 -0800, smnewberger@comcast.net (Stuart M
> Newberger) in comp.ai.philosophy wrote:
>
> [. . .]
>
> > To Lester:The historical confusions of ordinary language is at
> >its best in discusions in older texts of ordinals and cardinals.
> > Lets stick to finite sets first.The natural numbers have a
> >natural order; 1<2<3<4..(etc,uh oh the infinite is starting to appear)
> >.An ordinal number is a natural number assigned to an ordered set
> >,that is,a (finite) set together with a given order,Like
> > dog<cat<mouse.We can match this set with the beginning of the
> >natural numbers (starting with 1 and leaveng none out)preserving the
> >order in exactly one way,namely:dog 1st,cat 2nd,mouse 3rd.The last
> >number that we use ,namely 3 is called the ordinal number of the
> >ordered set dog<cat<mouse.
> > So an ordinal number is just a number,not a special new kind of
> >number.
> >
> > There are 5 other ways of ordering the set consisting dog
> > cat and
> >mouse
> > and thus 5 other ways of counting the set,namely
> >dog1st,mouse2nd,cat3rd ;cat1st,dog2nd,mouse3rd ;cat1st,mouse2nd,dog3rd
> >;mouse1st,dog2nd,cat3rd ;and finally mouse1st,cat2nd,dog3rd.There are
> >no other ways and you always get the same "ordinal number" 3 for each
> >ordering.This common
> >number 3 which you get whenever you count this set{cat
> > mouse dog} in any
> >order is called the cardinality or "cardinal number" of this
> >set.Namely 3.So cardinals and ordinals for finite sets turnout to be
> >the same-namely just the number of elements in the set,which is
> >independent of who does the counting-thats important.So at least
> >forget about the distance betweeen the ordinal3 and the cardinal 3,I
> >assure you it is 0.Ah,zero,now there is another story.
>
> Hi Stuart -
>
> I appreciate your taking the time to explain contemporary mathematical
> thinking on the subject of ordinals and cardinals. (Although I have to
> add that your typographical conventions make the text very difficult
> to read. A space following punctuation marks would make the text a lot
> easier to parse.)
>
> Let me see now if I can explain certain caveats to what you describe.
> For any set such as A(dog, cat, mouse), A has both an ordinality and
> cardinality of 3 according to you because A contains three elements.
> However, when you say this you have implicitly converted the subject
> under discussion from set A(dog, cat, mouse) to A(element, element,
> element). This may be true but what are we then to make of the set
> A(dog, cat, mouse) which is the actual subject we are concerned with.
>
> There is still an order to A(dog, cat, mouse) but no cardinality. We
> can move among the elements by adding intervals to the beginning of A.
> For example, we can get to cat by adding dog to the beginning of A or
> to mouse by adding cat to that result, but we cannot get to cat by
> adding mouse to the start of A. Nor can we calculate the location of
> any element in the set through cardinal arithmetic operations. So, we
> cannot maintain that set A(dog, cat, mouse) has any cardinality
> whatsoever. Set A only has an ordinality associated with it.
>
> Basically the only sets with cardinality are those having all the same
> members, whether the members be arithmetic intervals 1, 2, 3, etc. or
> element, element, element where calculations are possible in cardinal
> arithmetic terms.
>
> Regards - Lester
Hello again.Sorry about my typography,I was hurrying too much.I think
your reply is showing some difficulties with the way mathemtics people
use language to describe things.I would write A={dog,cat,mouse}and
this statement means
that the letter "A" and the expresion "{dog,cat,mouse}" both denote
the same set,namely the one which has exactly the members shown in
the latter expression. When I say that they denote the same set I
mean only that they have the same members and nothing about any order
that I have written a list of the members (like in the expression to
the right of the equality).
So yes I am saying that {dog,cat,mouse}={cat,mouse,dog} ,that is the
expressions you see on either side of the = sign denote the same set
which is a mathematical object to be thought of as a collection of
things ,independent of the order of the nemes used to list them .The
names on the left and right are written in different order and the
convention is left first(left to right)but the set A
={dog,cat,mouse}={cat,mouse,dog} (three ways of denoting the same
set}does not have an order.This set has no distinguished order so it
has no ordinality.It is a finite set so it has a cardinality namely 3
because it has three members,thts all the cardinality is,-the number
of elements of the set.Then what has ordinality.The answer is a
composite object consisting of a set together with an assignment which
is first ,which is 2nd etc .I denoted this composite object previosly
by say{cat 1st,dog 2nd ,mouse 3rd}and this is a different mathematical
object then {dog 1st,cat 2nd,mouse 3rd} even though they both have
the same underlying set A.These sets have ordinality,3 because thats
how many of the natural numbers 1,2,3,...assigning thim from left to
right and not leaving any out ({cat 2,dog 5 ,mouse 3}is not allowed as
an ordered set. Previously I denoted the ordered set (cat 1,dog
2,mouse 3}by
{cat<dog<mouse} because there is another way of defining an ordering
"<" which works for infinite sets also.
Again no matter how you specify the order for a finite set A the
ordinality of the ordered set(the composite object) is 3.This common
number 3 is the cardinality of the set A.So for any finite ordered
set ,the ordinality is the same number as the cardinality of the set
being ordered.Regards,Stuart Newberger
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