Algebra Know-It-ALL

A definition for infinity

When we write out any particular natural number p in the form of a set of smaller numbers,
that set has exactly p elements in it:

0 =∅ which has no elements

1 = {0} which has one element

2 = {0, 1} which has two elements

3 = {0, 1, 2} which has three elements

↓

p= {0, 1, 2, 3, ..., p− 1} which has p elements

↓

and so on, forever

Now think of the set of all natural numbers. This set, which we called W (for whole numbers)
in the last chapter, is symbolized N. We can write

N= {0, 1, 2, 3, ...}

Think about this set N for a minute. What number does it represent, according to the scheme
we’ve just invented for building natural numbers? Can the set N represent any number at all?
We can never finish writing the list of its elements. But the mere fact that we can’t write the
whole list doesn’t mean that the set itself does not exist. We can imagine it, so in the math-
ematical world, it exists!
Mathematicians define the number represented by the entire set N as a form of “infinity”
and denote it using the last letter in the Greek alphabet, omega, in lowercase (ω). “Omega” is a
traditional expression for “the end of all things.” In formal terms, ω is called an infinite ordinal
ortransfinite ordinal, and it has some strange properties. There are infinitely many infinite
ordinals! We won’t delve into their properties here, but if you’re interested in learning more

0

1

Proceed forever!

2

{0}

{0, 1}

{0, 1, 2}

{0, 1, 2, 3}

5 {0, 1, 2, 3, 4}

4

3

Figure 3-3 The natural numbers

can be built up, each

one “on top” of its

predecessors.

How Natural Numbers are Made 37