Cardinal and Ordinal Numbers

Date: 01/08/97 at 16:12:21
From: Rebecca Aitchison
Subject: Cardinal/Ordinal Numbers

How can the number three (3) be both a cardinal and ordinal number at 
the same time?

Date: 01/08/97 at 18:19:46
From: Doctor Tom
Subject: Re: Cardinal/Ordinal Numbers

Hi Rebecca,

I remember when I was your age being quite confused by the difference 
between cardinal numbers and ordinal numbers.  Now I realize that it 
was probably because my teachers were confused as well!

When I was your age, the best explanation I got was something muddled 
like, "three" is a cardinal; "third" is an ordinal.

The reason for all the confusion is that the difference is a fairly 
deep subject in mathematics, but I'll try to explain it.

In fact, from a mathematician's point of view, all the non-negative 
integers: 0, 1, 2, 3, 4, ... are both cardinal and ordinal numbers.  
There is only a difference when you begin to look at "infinite" 
cardinals and ordinals.

Cardinal numbers are for counting things.  If you have two sets, and 
want to find out if there are the same number of things in each, you 
can try to match them up, and if you can match everything in the first 
set with something in the second so that all the things in both sets 
are used up, the sets have the same number of things, right?

For example, the sets {1, 4, 7, 17} and { A, 43, pig, elephant} are 
the same size because I can make the following matchings:

 1 <=> A
 4 <=> 43
 7 <=> pig
17 <=> elephant

Cardinal numbers are usually defined to be special sets that you can 
use to try to match with others.  Here is the usual definition:

0 = {  }  -- the empty set
1 = { 0 }
2 = { 0, 1 }
3 = { 0, 1, 2 }
4 = { 0, 1, 2, 3 }
and so on.

Notice that if you can match the set we called "4" with another set, 
that other set has 4 elements, so these cardinal numbers are great for 
counting things.

Notice the following pattern:

0 = { }
1 = 0 union { 0 }
2 = 1 union { 1 }
3 = 2 union { 2 }
and so on.

So n+1 = n union { n }.

These finite cardinals are also great for ordering - 0 is the first 
number, 1 is the second, 2 the third, and so on.

All the finite cardinals are ordinals as well.

Where you get into trouble is when you hit the first infinite 
cardinal.  How big is this set:

   { 0, 1, 2, 3, 4, 5, 6, ... }?

In other words, how big is the set that contains all the finite 
ordinals?  It's clearly infinite, because you can't match it up with 
any of the finite cardinals - there will always be numbers left over 
in the infinte set, no matter how large an ordinal you choose.

So mathematicans call this the first infinite cardinal, and usually 
write it as aleph-0 ("aleph" is the first letter of the Hebrew 
alphabet, and was used by George Cantor, a Jew who was the first to 
discover the transfinite numbers).

So after aleph-0, what's the next ordinal?  Well the usual way to get 
to the next one is this:

aleph-0 + 1 = aleph-0 union { aleph-0 }.

Or I can write it this way:

{ aleph-0, 0, 1, 2, 3, 4, ... }.

But I can match this up with aleph-0 as follows:

aleph-0 <=> 0
0 <=> 1
1 <=> 2
2 <=> 3
et cetera.

So aleph-0 and aleph-0 + 1 are the same size.  They have the same 
cardinality, but they are clearly different ordinals.

I won't go into it but there are also larger cardinals.  In 
particular, the size of the real numbers is larger than aleph-0.

If you're interested why, find a book that talks about transfinite 
numbers, and about Cantor's diagonalization proof.

You can learn more about transfinite numbers at:

http://mathforum.org/dr.math/problems/adams5.28.96.html   

You can read more about Cantor's diagonalization proof at:

http://mathforum.org/dr.math/problems/klimkow.4.8.97.html   

Pretty complicated, huh?  No wonder it's confusing.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/