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More numbers, or more squares? 0:00
Here's a puzzle. Which are there more of: the counting numbers, one, two, three, forever, or the square numbers, one, four, nine, sixteen? Surely the counting numbers, because most of them aren't squares. But pair them up: one with one, two with four, three with nine. Every number gets exactly one square, and nothing is left over. So which is it? And can one infinity be bigger than another?
Two ways to say "as many" 0:32
How do you know two groups are the same size? You could count them. Or pair them off, a cup on every saucer, and if nothing's left over, they match. We trust a second rule just as much: a part is smaller than the whole, an idea that goes back to Euclid. For anything you can finish counting, the two rules always agree. The puzzle is what happens when they don't.
Aristotle's way out 1:00
The most influential answer came from Aristotle, in Greece in the fourth century BC. He wasn't the first to worry about infinity, but his answer shaped the next two thousand years. The infinite, he said, never exists all at once. You can always count one more, or halve a line again, but at every stage you still have only finitely many. In a sense, he wrote, the infinite is, and in a sense it is not.
Potential and actual 1:30
That's potential infinity: a process that never stops, with always a next number. Its opposite is actual infinity: the whole endless collection, finished and held at once, every counting number in one bag. Even Euclid stated his result about primes the potential way: there are more primes than any number of them you care to name. As late as 1831, Carl Friedrich Gauss protested against treating the infinite as something completed. But the squares puzzle needs the whole bag.
A book written under guard 2:05
In 1638 Galileo Galilei was an old man under house arrest near Florence, watched by the Inquisition. That year his book Two New Sciences was printed in Leiden. In it, a character called Salviati walks two companions through the squares. Up to a hundred, one number in ten is a square; up to a million, one in a thousand. The squares get rarer and rarer, yet they still pair off with all the numbers.
Galileo's verdict 2:40
Galileo's verdict was to stop comparing. Equal, greater and less, says Salviati, are not applicable to infinite quantities. He wasn't the first to hit this wall. Nicole Oresme and Albert of Saxony argued over similar puzzles in the 1300s, and Archimedes may have called two infinite collections equal in number. But Galileo's famous answer amounts to: don't ask.
Pick one rule 3:10
The other way out: for infinite collections you can't keep both rules, so pick one. In a book published after his death, in 1851, Bernard Bolzano paired every point of a line five long with every point of a line twelve long, then refused to call them equally many. He kept the part-whole rule. The bolder choice is to trust the pairing, and the clearest picture of that choice is a hotel.
The hotel that is always full 3:41
Imagine a hotel with infinitely many rooms, numbered one, two, three, forever, and every room is taken. A new guest arrives. Easy: everyone moves up one room, and room one is free. There's no last room, so nobody falls off the end. Now a coach brings infinitely many new guests. Everyone moves to double their room number, one to two, two to four, three to six, and every odd-numbered room is empty. A full hotel can still find room for more.
Whose hotel? 4:18
It's called Hilbert's hotel, but for decades nobody knew whether David Hilbert had ever told it. The answer turned up in notes of his lectures in Göttingen in the winter of 1924 to 1925, never published in his lifetime and only printed in 2013. The story only spread after the physicist George Gamow put it in a popular book in 1947. In 2014 the historian Helge Kragh, who had once suggested Gamow invented it, wrote up the real story and publicly corrected himself.
A question for Dedekind 4:55
Long before that, the man who took pairing seriously was Georg Cantor, at the University of Halle in Germany. On the twenty-ninth of November 1873 he wrote to his friend Richard Dedekind. Could the counting numbers be paired, one to one, with all the positive real numbers? Dedekind couldn't answer, though he sent back a proof Cantor later used without credit, and said the question had no particular practical interest. Eight days later Cantor had a proof that the answer is no, published in 1874.
Same size, by definition 5:35
Cantor's key move was a definition. Two collections are the same size, in his word they have the same power, if they can be paired one to one with nothing left over. By that rule the squares and the counting numbers are the same size, and so are the even numbers. Even the negatives fit, if you list zero, one, minus one, two, minus two, and so on. Anything you can list like that, first, second, third, without missing one, is called countable.
Even the fractions 6:08
Fractions look harder. Between any two there's always another, so they seem to crowd the number line far more thickly. Yet Cantor showed in 1873 that they can be listed too. One way, the picture used in textbooks today: lay the positive fractions out in a grid, and snake through it one diagonal at a time, skipping repeats like two over two. Every fraction turns up somewhere on the list. So crowded isn't the same as bigger.
The number that isn't on the list 6:41
Now try the decimals between zero and one. Suppose someone hands you a list that claims to hold every one of them. Build a new decimal: make its first digit differ from the first number's first digit, its second from the second number's second digit, and so on down the diagonal; say, write a five, or a four if the digit was already five. Your new number can't be the first on the list; it differs in the first place. It can't be the second; it differs in the second place. It differs from every number on the list somewhere, so the list was never complete.
Letters, not digits 7:21
That's Cantor's diagonal argument, though it wasn't his first proof. In 1874 he'd used a more roundabout argument with intervals nested inside intervals. The diagonal came in a short paper in 1891. And Cantor didn't use decimals: his lists were endless strings of two letters, m and w. That showed it isn't really about numbers at all: it's a fact about endless lists themselves.
Why not just add it? 7:50
Why not just add the missing number to the list? You can, but that's a new list, and the same recipe finds a number it misses. The argument beats every possible list, so the decimals can't be listed at all: they're uncountable. That gives "bigger" its careful meaning. The counting numbers pair with part of the decimals, one with a tenth, two with a hundredth, three with a thousandth. But no pairing ever uses up all the decimals. That's exactly what it means for one infinite collection to be bigger than another.
No biggest infinity 8:26
In the same paper, Cantor turned the trick on any collection at all. Look at all the ways of choosing some of its members. From three things there are eight ways to choose, counting none and all three. For any collection, finite or infinite, the choices can never be paired off with the members: there's always a choice left over. So from any infinity you can build a bigger one, and a bigger one again. There is no largest infinity.
A hundred years too soon 8:56
Many mathematicians resisted. Leopold Kronecker, in Berlin, held that maths should deal only with things built from whole numbers in finitely many steps. He's remembered for saying that God created the integers, and all else is the work of man. In 1877 he tried to stop one of Cantor's papers being published. And in 1885 even a friendly editor, Gösta Mittag-Leffler, talked Cantor into withdrawing a paper, calling it about a hundred years too soon.
Paradise 9:34
From 1884 Cantor suffered bouts of serious depression. It's often said the fight over infinity broke him, but historians argue his illness magnified those battles, rather than the battles causing his illness. By 1897, leading mathematicians were praising his ideas in public, and he died in Halle in 1918. In 1925 Hilbert told a meeting in Münster: no one shall expel us from the paradise which Cantor has created for us.
The gap in between 10:09
Cantor left one question he never settled. Is there an infinite size strictly between the counting numbers and the decimals? He guessed no; that guess is the continuum hypothesis. In 1900 Hilbert put it first on his famous list of problems. Then in 1938 Kurt Gödel showed the standard rules of set theory can't disprove it, and in 1963 Paul Cohen showed they can't prove it either, as long as those rules don't contradict themselves.
So what is infinity? 10:43
Mathematicians still argue over whether new axioms could settle it, or whether it has no single answer. So what is infinity? Not a number you reach by counting, and not twelve divided by zero. It describes collections that never run out, and when you compare them by pairing, some really are bigger than others. But the diagonal hides a snag. Some numbers have two decimal names: one tenth is zero point one, and also zero point zero nine nine nine, forever. That's why the recipe only ever writes fours and fives. Which raises a question. Does zero point nine nine nine, forever, really equal one?
Sources
Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.
- Galileo Galilei, Dialogues Concerning Two New Sciences, trans. H. Crew and A. de… — archive.org
- K. Easwaran, A. Hájek, P. Mancosu and G. Oppy, "Infinity", Stanford Encyclopedia of… — plato.stanford.edu
- E. Lehman, F. T. Leighton and A. R. Meyer, Mathematics for Computer Science, MIT… — courses.csail.mit.edu
- J. J. O'Connor and E. F. Robertson, "Infinity", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
- J. J. O'Connor and E. F. Robertson, "Aristotle", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
- J. J. O'Connor and E. F. Robertson, "Galileo Galilei", MacTutor History of… — mathshistory.st-andrews.ac.uk
- Library of Congress, title page of Discorsi e dimostrazioni matematiche intorno a due… — loc.gov
- José Ferreirós, "The Early Development of Set Theory", Stanford Encyclopedia of… — plato.stanford.edu
- Helge Kragh, "The True (?) Story of Hilbert's Infinite Hotel", arXiv:1403.0059 (2014) — arxiv.org
- J. J. O'Connor and E. F. Robertson, "Georg Ferdinand Ludwig Philipp Cantor", MacTutor… — mathshistory.st-andrews.ac.uk
- Robert Gray, "Georg Cantor and Transcendental Numbers", American Mathematical Monthly… — web.archive.org
- J. J. O'Connor and E. F. Robertson, "A history of set theory", MacTutor History of… — mathshistory.st-andrews.ac.uk
- J. J. O'Connor and E. F. Robertson, "Leopold Kronecker", MacTutor History of… — mathshistory.st-andrews.ac.uk
- Peter Koellner, "The Continuum Hypothesis", Stanford Encyclopedia of Philosophy (2013) — plato.stanford.edu
Image credits
- After Justus Sustermans (or studio), Galileo Galilei, c. 1640 · National Maritime Museum, Greenwich, Caird Collection · public domain (PD-Art) · via Wikimedia Commons
- Title page, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden 1638 · Library of Congress · public domain · via Wikimedia Commons
- David Hilbert, c. 1912 (Göttingen faculty postcard) · photographer unknown · public domain · via Wikimedia Commons
- Georg Cantor, carte de visite, studio Anders-Paltzow / Zeth, 1870–1885 · Staatliche Museen zu Berlin, Kunstbibliothek · public domain · via Wikimedia Commons · cropped
Researched and scripted with AI assistance, fact-checked claim by claim, with synthetic narration and diagrams drawn in code. How we make episodes.
