What is infinity? Why some infinities are bigger than others

Infinity is the idea of something without any end or limit. In mathematics it shows up in two ways: as a process that never stops, like counting 1, 2, 3 forever, and as the size of a collection that has no last member, like the set of all counting numbers. Infinity is not an ordinary number you can add or divide, and in the 1870s and 1890s Georg Cantor proved something stranger: some infinite collections are strictly bigger than others.

In short

• Infinity is not a very large number; it is the size of collections that never end, or the destination of processes that never stop.
• Two infinite sets are the same size if their members can be paired off one to one, so there are as many even numbers as counting numbers.
• Cantor's diagonal argument shows there are more real numbers than counting numbers, and there is no largest infinity.

Is infinity a number?

Not in ordinary arithmetic. If ∞ behaved like a number, ∞ + 1 = ∞ would mean, after subtracting ∞ from both sides, that 1 = 0. So expressions like ∞ − ∞ or ∞ ÷ ∞ are left undefined. The symbol ∞ itself was introduced by the English mathematician John Wallis in 1655.

The Greek philosopher Aristotle drew the distinction that still organizes the subject. A potential infinity is a process that can always go one step further: whatever number you name, you can add 1. An actual infinity is a completed whole with infinitely many parts, all at once. Aristotle accepted the first and rejected the second, and most mathematicians followed him for two thousand years. Cantor's work at the end of the nineteenth century made actual infinities precise, and they are now an ordinary part of mathematics. Unlike the numbers on the number line, Cantor's infinite numbers measure how many things a set holds.

How can two infinite sets be the same size?

For finite sets you can count. For infinite ones, Cantor used an older idea: two sets are the same size if their members can be paired off exactly, each with one partner, none left over. You can tell a theater is exactly full without counting, if every seat has one person and every person a seat.

Pairing gives results that look wrong at first. Match each counting number with its double: 1 with 2, 2 with 4, 3 with 6, and so on. Every counting number gets exactly one even partner, and every even number gets exactly one counting-number partner. So there are as many even numbers as counting numbers, even though the evens are only part of them. Galileo noticed the same thing with square numbers in 1638 and concluded that "equal", "greater" and "less" don't apply to infinite quantities. Cantor's answer was the reverse: for infinite sets, a part can be the same size as the whole, and that is practically the definition of being infinite.

Hilbert's hotel, room by room

The mathematician David Hilbert dramatized this in a lecture in 1924–1925, and the physicist George Gamow made the story famous in 1947. A hotel has infinitely many rooms, numbered 1, 2, 3 and on forever, and every room is occupied.

1. One new guest arrives. The manager asks every guest to move from room n to room n + 1: room 1 to 2, room 2 to 3, room 1,000 to 1,001. Everyone still has a room, and room 1 is free. The full hotel took one more.
2. A bus with infinitely many new guests arrives. The manager asks every guest to move from room n to room 2n: room 1 to 2, room 2 to 4, room 3 to 6. All current guests are now in even-numbered rooms, and the infinitely many odd rooms, 1, 3, 5, 7…, are free for the bus.
3. Infinitely many buses arrive, each with infinitely many passengers. Even this fits. One method uses prime numbers: move each current guest from room n to room 2ⁿ; give the passenger in seat s of the first bus room 3ˢ, of the second bus room 5ˢ, of the third 7ˢ, and so on through the odd primes. Because every whole number has only one prime factorization, no two people are ever sent to the same room.

The hotel is never "more full". The set of counting numbers is the smallest infinity, and its size has a name: aleph-null, written ℵ₀. Any set whose members can be listed in order, first, second, third, is called countable and has this size. Even the fractions are countable: write them in a grid by numerator and denominator and zigzag through it, skipping repeats.

Why are there more real numbers than counting numbers?

The real numbers, every point on the number line including the irrational ones like √2 and pi, cannot be listed. Cantor gave a first proof in 1874 and his most famous one, the diagonal argument, in 1891. It works by showing that any list of them must leave some out.

Suppose someone claims to have listed every real number between 0 and 1, as infinite decimals. Here is the start of such a list:

Position	Number on the list
1st	0.14159…
2nd	0.50000…
3rd	0.33333…
4th	0.71828…
5th	0.41421…

Now build a new number one digit at a time. Look at the bold digit on the diagonal: the 1st digit of the 1st number, the 2nd digit of the 2nd, and so on. Use a simple rule: if that digit is 5, write 4; otherwise, write 5.

• 1st digit of the 1st number is 1, so write 5.
• 2nd digit of the 2nd number is 0, so write 5.
• 3rd digit of the 3rd number is 3, so write 5.
• 4th digit of the 4th number is 2, so write 5.
• 5th digit of the 5th number is 1, so write 5.

The new number starts 0.55555… and continues by the same rule down the whole list. It differs from the 1st number in the 1st digit, from the 2nd in the 2nd digit, and from the nth number in the nth digit. So it is not anywhere on the list. (Using only the digits 4 and 5 avoids the one trap, that 0.4999… and 0.5000… are the same number.) The same move defeats any list anyone could make, so the real numbers are uncountable: a strictly bigger infinity than ℵ₀. The method, assume a complete list and build something it misses, is a kind of proof by contradiction.

It doesn't stop there. Cantor's theorem shows that the collection of all subsets of any set is always bigger than the set itself, so there is an endless ladder of ever-larger infinities and no largest one.

Still argued

Is there an infinity between the counting numbers and the real numbers? Cantor guessed not, the continuum hypothesis, and could not prove it. Hilbert put it first on his famous 1900 list of open problems. Kurt Gödel in 1938 and Paul Cohen in 1963 showed together that it can be neither proved nor disproved from the standard axioms of set theory. As the Stanford Encyclopedia of Philosophy describes, mathematicians still disagree about what that means: some, like Cohen, held that the question then has no answer; others, like Gödel, held that it has a true answer that new axioms could reveal.

Why does 0.999… equal 1?

Many people feel 0.999…, with nines forever, must fall just short of 1. It doesn't fall short at all; it is exactly 1. The confusion comes from treating the endless decimal as a process that is still going on. Mathematicians define it as the limit of the sequence 0.9, 0.99, 0.999 and so on, the single value those numbers close in on.

• The gap between 1 and 0.9 is 0.1; between 1 and 0.99 it is 0.01; after n nines it is 1/10ⁿ.
• Those gaps shrink below any positive number you could name, however small. So the limit has no gap at all: 0.999… = 1.
• A quick check by algebra: let x = 0.999… Then 10x = 9.999… Subtract: 10x − x = 9.999… − 0.999…, so 9x = 9, and x = 1.
• Another: 1/3 = 0.333…, and three times that is 0.999…, while three times 1/3 is 1.

This idea of a limit, an infinite process with a definite, finite result, is the foundation of calculus. It is also how mathematics answered Zeno of Elea's ancient puzzles: a runner covering half the distance, then half the rest, then half again, passes through infinitely many steps, but the distances 1/2 + 1/4 + 1/8 + … add up to exactly 1, and the race ends. Such puzzles are famous examples of what a paradox can teach.

Does infinity exist in the physical world?

Nobody knows. Whether space goes on forever is an open question: the part of the universe that can be observed is finite, and measurements so far are consistent with a universe that is either infinite or far bigger than what we see.

In physics, an infinite answer usually means a theory has been pushed past where it works. Around 1900, classical physics predicted that a hot object should radiate an infinite amount of energy at short wavelengths, the so-called ultraviolet catastrophe; Max Planck's idea that energy comes in small packets removed the infinity and began quantum theory. General relativity predicts infinite density at the center of a black hole and at the start of the Big Bang. Most physicists read those singularities not as real infinities but as signs that a theory combining gravity and quantum mechanics is still missing.

Check yourself

• Without looking back, explain how a full Hilbert's hotel fits a bus with infinitely many new guests.
• Walk through the diagonal argument and say why the new number can't be anywhere on the list.
• Why is 0.999… exactly equal to 1, rather than just very close?

Sources

• Infinity — Wikipedia
• Cantor's diagonal argument — Wikipedia
• Hilbert Hotel — Eric Weisstein, Wolfram MathWorld
• The Continuum Hypothesis — Peter Koellner, Stanford Encyclopedia of Philosophy
• Infinity — MacTutor History of Mathematics, University of St Andrews