# Beyond infinity

#### Expert reviewers

## Essentials

- Sets that have the same size as the set of natural numbers are called ‘countably infinite’. Examples include the set of even numbers and the set of rational numbers (numbers that can be written as fractions).
- The set of real numbers (numbers that live on the number line) is the first example of a set that is larger than the set of natural numbers—it is ‘uncountably infinite’.
- There is more than one ‘infinity’—in fact, there are infinitely-many infinities, each one larger than before!

Previously, we looked at methods of comparing the sizes of sets and discovered that many infinite sets could be paired up exactly, such as the set of
natural numbers
**GLOSSARY**
**natural numbers**The set of all positive whole numbers, which can be used to count individual objects. Depending on which mathematician you ask, this set may or may not include the number 0. Examples: 1, 2, 3, …
and the set of even numbers. These sets both have the same size, or ‘cardinality’—namely, ℵ_{0}, our familiar ‘infinity’.

Does every infinite set have the same size? Can they all be matched up exactly?

For a long time, it seemed as though all infinite sets would be able to be paired up with each other—meaning that every infinite set had the same size, ℵ_{0}. This intuitive idea was shattered in 1874 by the discovery of larger and larger infinities. In a five-page paper that shook the mathematical world (and gave rise to set theory in the process), the German mathematician Georg Cantor laid out his discovery of new horizons ‘beyond infinity’.

How is it possible to construct a ‘larger’ infinite set than the set of all natural numbers? Clearly, we’re going to need to add extra ingredients to the mix.

## Rational thinking

The first way we can expand our number system is by introducing all the negative numbers, forming the set of
integers
**GLOSSARY**
**integers**The set of all whole numbers, both positive and negative. The number 0 is definitely a part of this set. The natural numbers are a part of the integers. Examples: -2,-1,0,1,2, …
, ℤ. However, it soon becomes clear that this won’t help us reach a larger cardinality—we’ve effectively doubled the set ℕ by adding in a negative number for each positive number, and we know from our experience with combining the odd and even numbers that doubling an infinite set is not going to change its cardinality.

The next thing we can try is to include all the numbers that can be written as fractions, creating the set of
rational numbers
**GLOSSARY**
**rational numbers**The set of all numbers, both positive and negative, that can be written as a fraction. Equivalently, the set of all numbers whose decimal expansions either terminate or eventually repeat. The natural numbers and the integers are a part of the rational numbers. Examples: -3, 1/3, 0.5, …
, ℚ. We seem to have an awful lot of numbers in ℚ—for starters, there’s infinitely-many fractions between any two whole numbers! Clearly, it’s going to be difficult to pair off the set of natural numbers:

$$\mathbb{N}\text{ = \{0, 1, 2, 3, 4, 5, 6, … \}}$$

with the set of rational numbers:

$$\mathbb{Q}\text{ = \{ 0,} \frac{1}{2},\frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, … , 1, \frac{11}{2}, \frac{11}{3} … \}$$

Difficult … but not impossible. Cantor showed how this could be done by thinking about things a little differently. The key is to realise that in order to pair off the rational numbers with the natural numbers, we just need to find a way to list the rational numbers in some sort of order. We can then pair the first rational number in our list with the natural number 1, the second rational in the list with 2, and so on.

On the surface, this seems a tall order—how can we can list the rational numbers, which include fractions, in an order that captures every single one? There’s no such thing as the smallest fraction bigger than 0, for starters! But by doing a few mental zigzags, it turns out that it is possible to make such a list.

To start, we notice that every fraction is made from a pair of natural numbers—one on the top of the fraction (the numerator) and one on the bottom (the denominator). There are two caveats. First, we cannot have 0 on the denominator of the fraction. Second, we can write the same fraction in infinitely-many different ways using different pairs of numbers (for example, ^{1}⁄_{2} is the same as ^{2}⁄_{4}, and ^{3}⁄_{6}, and so on).

Since every rational number can be thought of as a pair of natural numbers, we can draw up the following infinite table to capture every possible rational number:

We only want each fraction to appear once in the table, and at the moment this table captures each fraction over and over again, infinitely-many times. So we take care to eliminate all the duplicates by crossing out each entry in the table where the fraction can be simplified (all the entries where the numerator and denominator share a common factor are removed).

Having done that, we can now zigzag our way through the table to catch every single fraction, like so:

Then all we have to do then is list every fraction in the order we encountered it during the zigzag:

Natural number | Rational number |
---|---|

1 | 0 |

2 | 1 |

3 | ^{1}⁄_{2} |

4 | 2 |

5 | ^{1}⁄_{3} |

… | … |

And, just like that, we’ve made our bijection between the rational numbers ℚ and the natural numbers ℕ! We can check that every natural number has a partner (the list of fractions never ends) and that every rational number has a partner (every rational appears exactly once in the table, after we carefully remove the duplicates). Although we didn’t capture all the negative rational numbers in our mapping, the same idea can be easily modified to take them into account.

So it turns out that the set of rational numbers is countable too—it has cardinality ℵ_{0}. It may seem like we’ve run out of options; that there really is only one type of infinity. But Cantor had one final trick up his sleeve, which he published in his famous 1874 paper. There is one more source of numbers we can draw on.

## The real surprise

Apart from natural numbers and rational numbers, there is one more set of numbers that most people are familiar with—the set of
real numbers
**GLOSSARY**
**real numbers**The set of all numbers that live on the number line. The natural numbers, integers and rational numbers are a part of the real numbers—but the real numbers also include numbers that cannot be written as a fraction, like π and the square root of pi, whose decimal expansions never terminate or repeat. Examples: -2, 0, 1/4, √3, log 2, …
, ℝ. The set of real numbers includes all the whole numbers and all the fractions, as well as lots of other numbers that can’t be written as a fraction, like π (pi) and the square root of 2.

Cantor was able to prove that it was impossible to create a bijection between the set of real numbers ℝ and the set of natural numbers ℕ. No matter how hard you try, you’ll always end up using up all of the natural numbers and still having real numbers left over, unpaired. There are definitely more real numbers, in a mathematically precise way, than there are natural numbers.

To prove this, Cantor used an elegant mathematical technique known as ‘proof by contradiction’. Cantor began by assuming that you could create the bijection, listing all real numbers in some order, just like we did for the rational numbers, in order to match them with the natural numbers. He then showed that no matter how you had created the list, he’d be able to show you a real number that didn’t appear on your list, thus contradicting your result and proving it couldn’t be done.

Even more incredible, Cantor’s ‘diagonalisation argument’ doesn’t even need all the real numbers to be considered. It turns out that any piece of the number line, no matter how tiny, contains ‘more’ numbers in it than all of the natural numbers (and all of the rationals too)! To illustrate Cantor’s argument, we’ll just take the set of real numbers between 0 and 1 and show that you can’t possibly match them up with the set of all natural numbers.

Suppose that we’ve created our bijection, producing a list containing every single real number between 0 and 1 in some order. Cantor then hands us another real number between 0 and 1 that can’t possibly be on our list. To create this number, he takes the first decimal place of our first number on our list and adds 1 to it, and uses the result as the first decimal place of his new number. (If the first decimal place of our first number was 9, he changes it to a 0). He then takes the second decimal place of our second number and adds 1 to it, using the result as the second decimal place of his new number. This process continues, resulting in a real number that Cantor claims won’t be on our list.

### Some infinities are greater than others

No matter how you pair them together, you can always find a real number that isn’t paired with a natural number. This means there are more real than natural numbers—even though there’s an infinite amount of both!

Use this interactive to pair up some natural numbers with the beginning of real numbers. We’ll always be able to generate a new number that’s not on your list. We do this by taking one digit from each real number (in a diagonal pattern) and incrementing it by one.

Natural number | Real number |
---|---|

1 | 0.01234 |

2 | 0.12345 |

3 | 0.23456 |

4 | 0.34567 |

5 | 0.45678 |

- Unmapped real number
- 0.13579

Natural number | Real number |
---|---|

1 | 0.10000 |

2 | 0.01000 |

3 | 0.00100 |

4 | 0.00010 |

5 | 0.00001 |

- Unmapped real number
- 0.22222

Natural number | Real number |
---|---|

1 | 0.22222 |

2 | 0.02000 |

3 | 0.00200 |

4 | 0.00020 |

5 | 0.00002 |

- Unmapped real number
- 0.33333

Natural number | Real number |
---|---|

1 | 0.13254 |

2 | 0.24365 |

3 | 0.35476 |

4 | 0.46586 |

5 | 0.57698 |

- Unmapped real number
- 0.25599

- Unmapped real number
- 0.00000
- Insert digits in each section then press ‘update result’

Is he right? Well, let’s try and find Cantor’s number in our list. It can’t be the first number, because the first decimal place (at least) doesn’t match up. It can’t be the second number on our list, because the second decimal place doesn’t match. It can’t be the millionth number on our list, because the millionth decimal place doesn’t match, and so on. No matter how we go about making the list, we can’t even capture every real number between 0 and 1!

Cantor’s argument works for the real numbers because the decimal expansion of a real number doesn’t have to have a pattern to it. This is in contrast to the decimal expansions of rational numbers, which must eventually repeat themselves.

So at last we have finally found a larger infinity than ℵ_{0}! Perhaps not surprisingly, this new infinity—the cardinality of the set of real numbers ℝ—is called ℵ_{1}. It’s the second transfinite cardinal number, and our first example of a bigger infinity than the ℵ_{0} infinity we know and love.

## Conclusion

So although Buzz Lightyear’s goals of travelling ‘to infinity and beyond’ doesn’t make sense in the literal sense—infinity is not a place, nor a destination, nor a final point—you could argue that mathematics allows us to explore infinity and beyond. Different infinite sets can have different cardinalities, and some are larger than others. Beyond the infinity known as ℵ_{0} (the cardinality of the natural numbers) there is ℵ_{1} (which is larger) … ℵ_{2} (which is larger still) … and, in fact, an infinite variety of different infinities.

It may seem esoteric, but the understanding of infinity—and set theory—is vital to understanding the very foundations of mathematics. It’s knowledge that was hard won over millennia, and that knowledge provides us with tools for understanding very real complexities in the world around us.