Infinite and Transfinite Numbers

Date: 5/28/96 at 13:49:24
From: Richard Adams
Subject: What is difference between infinite and transfinite? 

Can anyone explain to me, in a simple way, what transfinite 
numbers are and how they're different from infinite numbers? 
I seem to recall reading that someone (Hilbert?), found a way to do 
math with infinites and transfinites - how does that work? How can 
you subtract an infinite number from another infinite number and 
get any sort of reasonable answer?

Thanks in advance! :)

--Richard E. Adams

Date: 6/14/96 at 01:12:05 
From: Dr. Tom
Subject: Re: What is difference between infinite and transfinite? 

Hi Richard,

The word "infinite" in mathematics is a little dangerous, since it is 
used in many ways. For example, I can ask, 

"What is the limit, as x -> infinity, of (x+1)/(x-1)?" 

In this case, I can translate the problem into a form that doesn't use 
the word "infinity" as follows:

"Find a limit L such that given any epsilon > 0, there exists an 
M > 0 so that if x > M, then |(x+1)/(x-1) - L| < epsilon."

The concept of infinity is also used to answer questions like, "What 
is the area of the entire plane?" Clearly, whatever finite number 
you name, you can show that the "area" of the plane must be bigger 
than that.

But there's another very precise use of infinity, and it has to do 
with counting.

The set {A, B, C} has three elements, and I can prove it by 
constructing a one-to-one correspondence with a set that's known to 
have 3 elements. In fact, often the natural numbers are defined as 
sets:

0 = {} (the empty set)
1 = {0}
2 = {0,1}
3 = {0,1,2}

and so on. In general,

n+1 = n union {n}

The nice thing about this is that I can show that a set has 5 elements 
if I can find a one-to-one mapping of the elements in the set with 
the elements in 5 (where we think of 5 as the set {0,1,2,3,4}).

So what is the set:

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

that is, the set containing all the natural numbers. This is usually 
called "aleph-nought" (written with the Hebrew letter aleph, with a 
subscript of 0. Since I can't type Hebrew letters on my computer, 
I'll call it A0. This set is clearly infinite - it can't be matched up in 
a one-to-one way with any of the finite numbers.

So what's A0+1? Well, why not do the same thing: 

A0+1 = A0 union {A0}

In that way, we can get A0+2, ..., and if we take the union of all of 
those, we get what is called A0*2, et cetera. 

This process of unioning can be continued, in principal, forever, 
and the class of "numbers" generated is called the class of 
(transfinite) ordinal numbers.

I could write about this for a long time, but let me make a few 
observations (some of which are not obvious): 

A0 and A0+1 are different, but they are the same size. There is a 
way to match up all the objects in A0 one-to-one with those in 
A0+1.

There are ordinals, larger than A0, that cannot be matched one-to-
one with the elements of A0. For example, the set consisting of all 
the subsets of A0 cannot be matched one-to-one with the elements 
of A0. Look up Cantor's famous "diagonalization" proof.

The first time an ordinal is bigger than all those below it, it's called 
a "cardinal" number. Cardinal numbers measure the size of sets; 
ordinals are more closely related to ordering. 

There are more real numbers, for example, than there are natural 
numbers. But the question of whether the cardinality of the reals is 
the first cardinal greater than the cardinal A0 is undecidable - it is 
not invalid to assume it is, and it is not invalid to assume it is not. 
Paul Cohen proved this back in the 1960s using a method called 
"forcing". 

Just as you know that certain operations do not make sense in the 
natural numbers (for example, you cannot always divide one by 
another and stay inside the system of whole numbers), there are 
operations that do not make sense in the transfinite ordinals. In 
fact, addition and multiplication there are not commutative, and 
subtraction and division do not always make sense.

-Doctor Tom, The Math Forum