Algebra Know-It-ALL

about them, there are plenty of Web sites you can explore. Go to your favorite search engine

and enter “infinite ordinal” or “transfinite ordinal” in the phrase-search mode.

Are you confused?

In the solution to Prob. 2 in Chap. 2, we saw how sets can be built on each other by “tacking on braces”

to the null set:

∅

{∅}

{{∅}}

{{{∅}}}

{{{{∅}}}}

{{{{{∅}}}}}

↓

and so on, forever

Why can’t we do that to construct the natural numbers? Well, we could, perhaps. But this method doesn’t

increase the number of elements in each set as the defined number gets bigger. Each of the above sets,

except for the first, contains one element. To make a good definition of the natural numbers, number

theorists want to have the sets contain more and more elements as the defined numbers get larger.

Here’s a challenge!

Write down the natural numbers 0 through 4 purely in terms of braces and the null set symbol, showing

how each number can be assembled from “multiple nothings.”

Solution

Start with 0, which is equal to ∅. It helps if you write the numbers as sets of smaller numbers and then

break those expressions down:

0 =∅

1 = {0} = {∅}

2 = {0, 1} = {∅, {∅}}

3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}

4 = {0, 1, 2, 3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}

Special Natural Numbers

You can classify natural numbers in various ways, just as you can classify people according to

blood type, postal zone of residence, country of residence, or even (maybe someday) planet of

residence. Here are a few of the most well-known types of natural numbers.

38 Natural Numbers and Integers