Cardinal and Ordinal NumbersDate: 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/ |
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