What Is Real · ages 9–12
Can you add to something endless?
There are endless numbers. Now double every one of them. Did you get more than endless?
Half of a room, then half of what is left, then half again. Zeno pointed out that a careful person would conclude you can never arrive — there is always a remaining step. It took roughly two thousand years to see clearly what was wrong with the argument, and the answer is about infinity rather than about walking.
The modern surprise is stranger. Compare all the counting numbers with all the even numbers. There seem to be half as many evens. But pair them up: 1 with 2, 2 with 4, 3 with 6, and so on forever. Nothing is left over on either side. So by the test that matters for matching sets, there are exactly as many evens as there are numbers. Half of an endless thing is all of it.
Then Cantor showed something that broke the picture completely: this trick does not always work. Some endless collections cannot be paired with the counting numbers no matter how cleverly you try, which means there really are larger and smaller infinities — endless, and yet not the same size. Nobody has ever found a physical infinity to check against. So this is a place where one of the most reliable things we do — mathematics — describes something we have never met.
Try it
Count the even numbers and the counting numbers. Pair each counting number with an even one — 1 with 2, 2 with 4, 3 with 6. Now: does one list run out first?
What do you think?
Do you think there is a right answer to this question?
Talk about it
- Is infinity a number or a direction?
- Can one infinity be bigger than another?
- Does anything infinite actually exist, or only in maths?
Yes, but… — the arguments people actually make
Infinity is just a very big number.
Then name one bigger number and subtract what you like. The whole point is that there is no last one — which is not the same as a very large one.
You can't double something that never ends.
You can double every part of it: map each number to twice itself. Every double is still on the list, which is the strange part — nothing was added.
This is just a game with symbols, not a fact about the world.
Possibly. But space and time both appear to have no edge, and you have to say what that means — and whether 'no edge' is the same as 'endless'.
What other philosophers think
- ZenoArgued that motion is impossible, since to cross a room you must first cross half, then half of the rest, forever. A paradox about infinity that nobody solved quickly.
- AristotleDistinguished a potential infinity — always able to go on — from a completed infinity that is all there at once. He accepted the first and not the second.
- GalileoNoticed that the squares of whole numbers can be matched one-to-one with all whole numbers, and concluded this cannot be done with finite quantities. He was close to a revolution.
- Georg CantorShowed that some infinities really are larger than others, and that this is provable rather than mystical.
If you want to go further
- Two New Sciences — Galileo Galilei1638. Where the one-to-one matching of infinite collections is first noticed.
- Contributions to the Founding of the Theory of Transfinite Numbers — Georg CantorThe work that made different sizes of infinity respectable and changed mathematics.
Most of these were written for adults, which is why it is worth reading them with someone older.