To Infinity and (slightly) Beyond

This article uses some relatively simple math, set theory, to show you how to add one to infinity and get something that you can see is different. The result is still an infinity, but a different one.

Introduction

The most effective technique for solving mathematical problems is to get your definitions right. Compare Roman to Arabic numerals for instance, and then think about which system you would prefer to do arithmetic in. The difference in the way numbers are represented gives the Arabic system a huge advantage.

In this article we are going to define a different way of representing numbers, one that looks substantially less efficient than even the Roman technique. This definition will however illustrate the mathematical principle of minimalism (doing the job with the smallest possible set of tools) and will, in the end, give us access to numbers we didn't even think were numbers before. In order to do this we need to define a few mathematical basics from set theory. If you already know them, just skim through the next section quickly.

A Little Set Theory

A set is a collection of objects, no two of which are the same. We write a set as a list of objects, in no particular order, separated by commas and enclosed in curly braces. For example {1,2,3} is a set containing the numbers 1, 2, and 3. The collection of objects {1,1, 2} is not a set because the number 1 appears twice.

The empty set is a set that has no members. We can write the empty set like this: {}. The empty set is a very odd set. It is, for example, the set of all human beings that weigh more than ten tons. The size or cardinality of a set is the number of elements it contains. The cardinality of the empty set is zero.

The union of two sets A and B is another set C which has, as members, every member of A and every member of B but no others. We write this mathematically as

C = A  ∪  B

Representing Whole Numbers as Sets

One way to define mathematics with the smallest number of parts is to start with set theory. Using only sets, we can create a representation of the positive whole numbers by representing the number n as a set with n members. We can even avoid using the symbols for digits if we are careful. Zero is represented by the empty set, which has zero members. One is represented by a set with one element. Since we've only defined one object so far, the number zero, that is the member of the set we use to represent one. Likewise, two is represented by a set with two elements: the numbers zero and one. The table below shows the numbers as sets of numbers, and then using only curly braces and commas.

Number Set Representation Curly Braces Only
0 {} {}
1 {0} {{}}
2 {0,1} {{},{{}}}
3 {0,1,2} {{},{{}},{{},{{}}}}
4 {0,1,2,3} {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}
5 {0,1,2,3,4} {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}}

The numbers from zero to five, represented with curly braces only, demonstrate beyond a shadow of a doubt that this is not a space-saving way to represent numbers. The only advantage, so far, is that we can represent all the positive whole numbers with the three symbols { , }.

A Brand New Number: ω

This notation gives us an additional capability. Each (whole, positive) number is represented by the set of all numbers that are at least zero but smaller than itself. This means that the set of all positive whole numbers:

ω = {0,1,2,3,4,5,...}

fits our definition of number. Now notice that if a number n, viewed as a set, contains another number m then we can see that n is larger than m. This means that ω is larger than all the (finite) positive whole numbers. This means that ω is an infinite number. In fact ω is the smallest infinite number.

Adding One to Numbers

Another property of the curly brace representation for whole numbers is that it permits us to use set theory to add one to a number. Looking at the table above, if we want to find a set of n+1 then we take the set for n and add n itself into it. For example:

2+1=3={0,1,2}={0,1} with 2 added in.
The process of adding in can be done using the union of sets.
2+1=2 ∪ {2}={0,1} ∪ {2}={0,1,2}=3

This means that, in general:

n+1=n ∪ {n}

Using this definition of adding one we see that

ω+1={0,1,2,3,4,5,...,ω}

Since every member of ω is finite and one member of ω+1 is infinite, they are clearly different infinite numbers. In fact we see that ω+1 is actually larger than ω because it contains it. The curly brace notation lets us do arithmetic, at least of a sort, on infinite numbers.

Things to Think About

  1. Are there other infinite numbers?
  2. Can you create any more arithmetic than adding one with the definitions above?
  3. What is the size (cardinality) of an infinite set?

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