Full transcript of the episode (it gives away every surprise)
Question for the reader
Are there more whole numbers (1, 2, 3, 4…) than even numbers (2, 4, 6, 8…)?
- More whole numbers, obviously
- Just as many (correct answer)
- The question makes no sense
If you answer “More whole numbers, obviously”:
There are exactly as many.
If you answer “Just as many”:
Yes, just as many. And it’s dizzying.
Pair each whole number with its double: 1 and 2, 2 and 4, 3 and 6… Every whole number finds an even partner, and every even number a whole partner. Nobody is left out, nobody is counted twice.
Interactive experiment
Galileo’s paradox
In 1638, Galileo noticed the same thing with squares: 1, 4, 9, 16… They become rarer and rarer, and yet they can be paired one by one with all the whole numbers. Check for yourself.
The whole numbers are paired one by one with their squares (1 with 1, 2 with 4, 3 with 9…). The further you go, the rarer the squares are among the whole numbers, and yet each one keeps its partner.
A part can be as big as the whole.
That is even the modern definition of an infinite set. Galileo, baffled, concluded that “bigger” and “equal” make no sense for infinity. He was wrong.
The Infinite Hotel
There are several infinities, and some are bigger than others. Some questions about them will never have an answer.
In 1924, the mathematician David Hilbert told a story to make people feel this strangeness: a hotel with infinitely many rooms, all of them taken.
Interactive experiment
Hilbert’s hotel
The hotel is full. A visitor arrives. Then a whole bus, with infinitely many passengers. Find them rooms.
In an infinite hotel that is full, a new guest finds a room if every occupant moves to the next room. A bus carrying infinitely many travelers finds room if every occupant moves to the room with double their number: all the odd-numbered rooms are freed up.
An infinite hotel that is full always has room.
Hilbert wanted to make people feel that infinity doesn’t behave like a very large number. And the strangest part is still to come.
Question for the reader
So are all infinities the same size?
- Yes: infinity is infinity
- No: some are bigger than others (correct answer)
If you answer “Yes: infinity is infinity”:
No. And it can be proved.
If you answer “No: some are bigger than others”:
Right. And it can be proved.
Interactive experiment
Cantor’s diagonal
Let’s imagine an infinite list that contains every number between 0 and 1. Let’s build a number that isn’t on it.
An infinite list of decimal numbers is written out. A new number is built by changing the first decimal place of the first, the second of the second, and so on: it differs from every number on the list, which therefore could not have been complete.
Some infinities are bigger than others.
Decimal numbers cannot be numbered off: there are strictly more of them than whole numbers. And above them, a ladder of ever larger infinities that never ends.
Georg Cantor proved it at the end of the 19th century. The discovery was so shocking that some colleagues called him a charlatan; Leopold Kronecker, one of the most influential, is said to have called him a “corrupter of youth.”
Cantor, worn down by the attacks and by depression, spent long stays in clinics. He died in 1918, poor and starving, in a sanatorium in Halle.
“No one shall expel us from the paradise that Cantor has created for us.”
Question for the reader
Is there an infinity bigger than that of the whole numbers, but smaller than that of the decimal numbers?
- Yes
- No
- We don’t know yet
- We can never know (correct answer)
If you answer “Yes”:
It’s worse than that: we can never know.
If you answer “We can never know”:
Exactly: we can never know.
This is the “continuum hypothesis,” posed by Cantor. In 1940, Kurt Gödel showed that it cannot be disproved with the usual axioms of mathematics. In 1963, Paul Cohen showed that it cannot be proved either. A perfectly clear question, which we know has no answer.
Mathematics has questions without answers. And it knows it.
Infinity isn’t just a game of the mind. Without it, there is no calculus, and so no modern physics, no engineering, no computing. Every time your phone works out a route, it handles limits: a tamed infinity.
Countable. Said of an infinite set whose elements can be numbered one by one, like the whole numbers, the even numbers or the fractions. The real numbers are not: their infinity is bigger.
Mathematicians tamed infinity on paper. But nature has even more baffling surprises in store. All you have to do is send light through two slits.
Sources
- Galileo Galilei, Two New Sciences, 1638.
- David Hilbert, “Über das Unendliche,” Mathematische Annalen, 1926.
- Georg Cantor, “Über eine elementare Frage der Mannigfaltigkeitslehre,” 1891; Joseph Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, 1979.
- Kurt Gödel, 1940; Paul Cohen, “The independence of the continuum hypothesis,” PNAS, 1963.