Since the time of Cantor, there have been two types of arithmetic, ordinal and cardinal. Ordinal arithmetic is generally
what students learn in elementary school and is the arithmetic that people think of when they hear the word arithmetic. Cardinal arithmetic deals with set theory
and comparing the size of sets. Each of these systems has its own operations of addition, multiplication and exponentiation (Birkhoff, 1942) 5 .
Cantor was the father of cardinal or transfinite arithmetic. He was the first to open his mind to understanding infinity and its different sizes. He needed a
language to describe his different infinities, so he decided to call them transfinite numbers. A transfinite number is a type of infinity. Cantor also wanted
a symbol to describe transfinite numbers so he decided to write them using ℵ, the Hebrew letter aleph. ℵ happens
to be the first letter in the Hebrew word for God. It is thought that Cantor chose this symbol because of his
religious beliefs. Cantor had a strong belief in God and thought that infinity was a way of representing Him because God is said to be infinite (Aczel, 2000) 1 .
Cantor believed that there was a series of alephs to describe higher and higher orders of infinity. For example, the
smallest or first order of infinity is countable infinity and is denoted ℵ 0 . This order includes the set of whole numbers, the set of
integers and the set of algebraic numbers. Cantor did arithmetic with the cardinality of these infinities. For example, ℵ 0 + 1 would be
like adding 1 to a countably infinite set. This doesn't change the size of the set, because it is still countably infinite (Aczel, 2000) 1 .
Other problems that Cantor examined were ℵ 0 + n (where n is an integer), ℵ 0 + ℵ 0 , ℵ 0
* n, ℵ 0 * ℵ 0 , and 2 ℵ 0 . With transfinite arithmetic, Cantor was trying to find an
operation that could take ℵ 0 to a higher order. This problem would become one that would haunt him for the rest of his life.
Use the applet below to visualize transfinite arithmetic. Read the questions to understand how the discrete points on the
graphs can be viewed as infinite sets.
As you might have surmised from the applet, Cantor discovered that adding anything to an infinity will never change its
size. Therefore,
ℵ 0 + n = ℵ 0 and
ℵ 0 + ℵ 0 = ℵ 0
Since multiplication is just repeated addition, it follows that
ℵ 0 * n = ℵ 0 and
ℵ 0 * ℵ 0 = ℵ 0
Cantor ran into trouble with transfinite arithmetic when he began to ponder exponentiation. He knew that given any finite set, the way to count the number of sets that could possibly
be made with the elements of that set was to take the power of 2 and raise it to the cardinality of the set. For example, if I have the set {1, 2}, the possible
sets that can be produced with the elements of that set are {1}, {2}, {1, 2} or {}. There are 4 possiblilities or 2 2 possibilities. Using this
logic, the number of sets that can be made out of ℵ 0 is found by 2 ℵ 0 . This calculation obviously produces
a much higher number then ℵ 0 , so Cantor thought that the solution to this problem must be a higher order of infinity, or ℵ 1 (Aczel, 2000) 1 .
Cantor's real question was if ℵ 1 was the continuum then were their any other infinities between ℵ 0 and ℵ 1 ?
Cantor spent the rest of his life trying to prove that 2 ℵ 0 = ℵ 1 , which implies that there are only 2
orders of infinity, countable infinity and the continuum. Unfortunately he was never able to prove or disprove the continuum hypothesis. His hypothesis has
been one of the most enduring problems of mathematics and remains a mystery today (Aczel, 2000) 1 .
Surreal Numbers
Listen to this podcast to learn a little bit more about transfinite arithmetic and discover surreal numbers.