Transfinite Numbers

Date: 11/07/97 at 00:45:00
From: John McSwain
Subject: Transfinite numbers

I have done some research and found that Georg Cantor discovered 
transfinite numbers, but I don't understand what they are. Could you 
please give an example or some depiction that might help me?

Thanks a lot.

Date: 11/07/97 at 10:39:22
From: Doctor Rob
Subject: Re: Transfinite numbers

To talk about cardinal numbers, including transfinite cardinal 
numbers, you need to talk about counting the numbers of elements in 
sets.

If a set A is finite, there is a nonnegative integer, denoted #A, 
which is the number of elements in A.  That is one of the finite 
cardinal numbers.

To do arithmetic with cardinal numbers, you use facts about finite 
sets and the number of elements in them, such as the following:

   If A and B can be put into one-to-one correspondence, then #A = #B,
      and conversely.
   If A is contained in B, then #A <= #B.
   If A is disjoint from B and C is their union, then #C = #A + #B.
   If A and B are sets, and C = A x B is the set of all ordered pairs
      of elements, the first from A and the second from B, then
      #C = (#A)*(#B).
   If C is the set of all subsets of A, then #C = 2^(#A).

Try some of these with small sets to satisfy yourself that they are 
so.

If the set is infinite, the corresponding cardinal number is not one 
of the finite cardinal numbers, so it is called a transfinite (or 
infinite) cardinal number.

The first transfinite number is called aleph-sub-zero (or aleph-
naught, or aleph_0). (Aleph is the first letter of the Hebrew 
alphabet.)  It is the cardinal number of the set of positive integers.  
Sets having this cardinal number are called countably infinite sets, 
or countable sets, because they can be put into one-to-one 
correspondence with the positive integers, or counting numbers.

The above rules for computing with finite cardinal numbers were 
extended by Cantor to apply to transfinite cardinal numbers. In this 
way he could talk about doing arithmetic with transfinite cardinal 
numbers. He discovered and proved that, if n is any finite cardinal 
number,

   aleph_0 + n = n + aleph_0 = aleph_0,
   aleph_0 + aleph_0 = aleph_0,
   aleph_0*n = n*aleph_0 = aleph_0   (n > 0),
   aleph_0*aleph_0 = aleph_0,
   aleph_0^n = aleph_0   (n > 0).

This is a pretty strange arithmetic! Note that subtraction and 
division are not definable operations in this arithmetic.  The 
Associative Laws of Addition and Multiplication hold, and the 
Commutative Laws of Addition and Multiplication do, too, and the 
Distributive Law, but there is no zero, no unity, no negatives, 
and no reciprocals.

Cantor also proved that 2^aleph_0 > aleph_0. This means that the 
set of all subsets of the integers cannot be put in one-to-one 
correspondence with the integers, that this set is really a different 
size of infinite set, truly larger. Cantor called 2^aleph_0 = C,
which stands for "continuum".  C is the cardinality of the set
of real numbers.  Once again, Cantor showed that

   n^aleph_0 = C   (n > 1),
   aleph_0^aleph_0 = C,
   C + n = n + C = C,
   C + aleph_0 = aleph_0 + C = C,
   C + C = C,,
   C*n = n*C = C   (n > 0),
   C*aleph_0 = aleph_0*C = C,
   C*C = C,
   C^n = C   (n > 0),
   C^aleph_0 = C,

but 2^C > C.  This trick of exponentiating one cardinal number to get 
a larger one works for all transfinite cardinal numbers. 

In this way, Cantor showed that there are infinitely many different
transfinite cardinal numbers. In fact the set of transfinite cardinal
numbers itself has a cardinal number, which is transfinite! Let's not
think about that - it gets very confusing!

But are there any transfinite cardinal numbers between aleph_0 and C?
The name aleph_1 has been given to the smallest transfinite cardinal 
number larger than aleph_0.  Then aleph_0 < aleph_1 <= C.  The above
question can be rephrased as, "Is aleph_1 = C?"  Cantor guessed that
this is so, but was unable to prove this.  It was one of the great 
unsolved problems of mathematics, called "The Continuum Hypothesis". 
Finally it was shown that this was not provable from the usual axioms
of set theory!  It is usually assumed as an additional axiom. 

I hope that this is helpful.  If you need more help, write again.

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