Building new numbers
Let’s make up a rule that we can use to generate new natural numbers, one after another.
Suppose we’ve built a certain natural number. Call it n. If we want to create the next higher
natural number, n+ 1, we can take all the natural numbers up to and including n, make them
into elements of a set, and then call that set the new number. A mathematician would write
it like this:
n+ 1 = {0, 1, 2, 3, ..., n}
If we have any particular natural number p, we can define it on the basis of all the natural
numbers less than itself, like this:
p= {0, 1, 2, 3, 4, ..., p − 1}
If that’s a little too abstract for you, look at Fig. 3-3. This drawing shows the first six natural
numbers (0, 1, 2, 3, 4, and 5) as they are built up and assigned to a vertical stack of points that
ascends upward as far as we care to go.
This is a nifty scheme! A natural number p is a set containing p elements. In theory, those
elements could be anything, such as apples, stars, atoms, or people. But it’s convenient to use
all the natural numbers less than p as those elements. That allows us to build the set of natural
numbers, one on top of the other, like stacking coins. Just as each coin in the stack rests on all
the coins below itself, every element p in the set of natural numbers “rests on” all the natural
numbers “below” itself. Once this process is defined, it sets in motion a mathematical “chain
reaction” that never ends.
36 Natural Numbers and Integers
Start here
and go on forever
Figure 3-1 The number 0 can be defined as the null set. We
can show how it starts to generate natural numbers
by placing it at the beginning of an endless string
of points.
{ }
(^01) Whole numbers
yet to be defined
Figure 3-2 The number 1 is the set containing the null
set, which is the same as the set containing the
number 0. It has one element.