Have you ever heard something described as “infinitely better” than something else? You know that’s impossible, right? That’s what infinity means – impossible to reach. You might also hear someone say they “gave 110%”. That’s impossible too.
Imprecision with numbers is one of my pet hates. It makes me mad. Do you know how mad it makes me? No, not infinitely mad. But really rather cross, nonetheless.
But there was one man who really did go mad because of infinity. His name was Georg Cantor.
Cantor was a 19th century mathematician, and he invented set theory, which is regarded as the foundation of modern mathematics. Without set theory you can’t really add, subtract, multiply or divide. You think you can, but that’s because you’ve learned cheats and shortcuts without really understanding what you were doing. After all, how do you “know” that 1 + 1 = 2? How do you even know what 1 and 2 mean? You were taught it in primary school, but that doesn’t mean you really understand what you’re doing.
Real mathematicians don’t talk about numbers. They talk about the cardinality of a set. That means the number of things in it. If a set doesn’t have any things in it (an empty set), its cardinality is zero. A set that contains the empty set has one item in it (the empty set). The set that contains this set and also the empty set has two items in it, and so on. This is how mathematicians define the cardinal numbers. (Aren’t mathematicians funny?)
Set theory is interesting (it is!), because it throws up bizarre paradoxes that seem to strike at the very heart of mathematics. If you always thought maths made no sense then you were right!
Here’s an example. Some sets are members of themselves. And some sets aren’t. For instance, the set of “mathematical ideas” contains itself. The set of “books” is not a book, so it doesn’t contain itself. Now, imagine the set of “sets that aren’t members of themselves.” Can you do this? I bet you can’t! But now ask: is this set a member of itself? Here’s the paradox. If this set is a member of itself, then it’s a set that isn’t a member of itself, so logically it can’t be in this set. And if the set isn’t a member of itself, then it’s not a “set that isn’t a member of itself” so logically it must be a member after all.
Don’t worry if this makes no sense to you. In more familiar terms, it’s the same logic that’s used to construct word paradoxes such as “This statement is false.” Intriguingly, these paradoxes lie right at the very core of mathematics and can’t be removed.
Anyway, this article isn’t about set theory. It’s about infinity. And set theory opens the door to infinity. In fact, when you open the door, infinity comes rushing through, in a scary and alarming way.
After inventing set theory, Cantor started to work on the concept of infinity, and suffered a series of mental breakdowns as a result. We’ve already seen how set theory can give you a nervous breakdown. How much more dangerous is it to use set theory to try to understand infinity? Answer: not infinitely more dangerous. But a lot.
Infinity is a bit like the sun. Mathematicians knew it was there, and they could glimpse it out of the corner of the eye, but they knew that if you looked at it directly it would burn your eyes. Cantor turned his gaze directly towards infinity – and beyond.
From the set theory definition of the cardinal numbers, we can imagine the set of all cardinal numbers. The number of cardinal numbers is infinite. Cantor called this number aleph zero, Aleph being the first letter of the Hebrew alphabet.
Aleph zero has unusual properties. Add 1 to aleph zero and the answer is aleph zero. Add aleph zero to itself and the answer is aleph zero. Multiply aleph zero by itself and the answer is still aleph zero. It seems that aleph zero is a limit that we can’t go beyond. Yet Cantor did.
Aleph zero has the property that it is countable. In other words, if you start counting the cardinality of an infinite set, it might take you forever, but at least there is a start point and a process – count the first number, count the second number, count the third number, etc. In principle, you could count all the elements of the set.
But are there sets that are uncountable? Cantor showed that the set of irrational numbers is uncountable. Irrational numbers are decimal numbers like pi that go on for ever (pi = 3.14159 …) The decimal bit just goes on and on, never repeating or coming to an end. How many numbers are there like this between 0 and 1? Well, lots. An infinite number. In fact, a number that is bigger than aleph zero.
A simple way of thinking about this is as follows. You can count whole numbers (1, 2, 3, …) because they are distinct and, well, countable. But between every pair of irrational numbers you can insert infinitely many more irrational numbers. For example, between 0.175… and 0.189… you can insert 0.177… and 0.181… and you can keep on inserting numbers indefinitely. That means that you absolutely cannot count them.
Mathematicians have shown that there is a hierarchy of infinities – aleph zero (infinite but countable), aleph one (the smallest uncountable infinity), aleph two, etc. Don’t ask me to explain what aleph two is, because I just don’t know.
Is your head hurting yet? Cantor’s was. After each breakthrough or setback he had another nervous breakdown, and it’s not surprising. Towards the end of his life, Cantor tried to prove that the number of irrational numbers between 0 and 1 was aleph one, but he was unable to. That was pretty much the last straw. Back to the clinic for poor, old Cantor. After his death in 1918, another mathematician, Paul Cohen, proved that it is impossible to prove whether or not the number of irrational numbers between 0 and 1 is aleph one. Bummer. Another mathematician, Kurt Gödel, proved that there are many theorems in mathematics that are unprovable – they literally cannot be proved or disproved. This relates back to the paradoxes of set theory that we explored earlier. How many unprovable theorems are there? That’s right – infinitely many. Gödel also suffered from mental illness, and died at the age of 52.
So if anyone tells you that mathematics is a gift from God, you might instead suggest that it is actually the work of the Devil.