Mathematics · sets, series, fractals
Infinity paradoxes that work
In a hotel with infinitely many rooms, even when every room is taken, there is always space for one more guest. Infinitely many steps can take a finite time, a curve with an infinite perimeter can enclose a finite area, and some infinities are bigger than others. Every paradox on this page can be explained: you will see what is merely surprising and what would be a genuine contradiction.
Laboratorium · Hilbert's Hotel
A full hotel, and still there's room
Every room has a guest. We don't throw anyone out — we just ask everyone to move to another room at the same time, following a simple rule.
Method for ∞ buses
View
The hotel is full: room n holds guest no. n. A new guest arrives — what will the receptionist do?
Who gets which room?
row = bus, column = seatThe zigzag runs along successive diagonals of the table. Every diagonal is finite, so every passenger on every bus eventually gets a room number — and no room is left empty.
Galileo's paradox (1638)
“Just as many” = pairs
Two sets have the same cardinality if their elements can be paired up so that nobody is left alone. Such a pairing is a bijection (a one-to-one correspondence). No counting needed — just match them up, like chairs and people in a room.
Countable sets
A set is countable if its elements can be numbered 1, 2, 3, … The cardinality of the natural numbers is aleph-null, ℵ0. The even numbers, the integers, the primes and — as you will see in the W tab — all fractions are countable.
What's the catch?
There is no contradiction: for finite hotels, “full” means “no vacancy.” For infinite ones that is simply false. Only our intuition, trained on a finite world, is surprised. A real hotel with infinitely many rooms can't be built, of course.
Laboratorium · countable and uncountable
Fractions can be numbered
Fractions are “dense”: between any two there is another. And yet all positive fractions can be lined up in a single sequence. The zigzag walks the diagonals of the grid and skips repeats (2/4 is the same as 1/2).
Zero and negative fractions fit too: 0, then alternately +a1, −a1, +a2, −a2, … So the rational numbers are countable, with cardinality ℵ0.
Cantor's diagonal: a number not on the list
list no. 1Suppose someone has made a list of all real numbers in the interval (0, 1). We build a number x: its k-th digit must be different from the k-th digit of the k-th row. Click a digit of x to change it, or use the rule: 5 → 4, anything else → 5.
Start with the first digit: look at the purple diagonal cells.
What just happened?
The number x differs from row 1 in place 1, from row 2 in place 2 — from every row k in place k. So it is not on the list. Adding it won't help: the new diagonal gives a new missing number.
Why don't we use the digits 0 and 9? Because 0.1999… = 0.2000… — a single number sometimes has two expansions. The digits 4 and 5 avoid that trap.
The screen shows only a few rows and digits. The argument works for the whole infinite list: row k always “breaks” at digit k.
The ladder of infinities
The power set P(A) is the set of all subsets of A. For n elements there are 2n of them. Choose n:
Cantor's theorem (1891): always |P(A)| > |A|, infinite sets included. If some function f assigned to each a a subset f(a) and used them all up, then the set B = {a : a ∉ f(a)} would be nobody's image — the diagonal in disguise. The ladder never ends:
The continuum hypothesis asks whether there is a cardinality between ℵ0 and 𝔠. Gödel (1938–1940) and Cohen (1963) showed that the standard axioms of set theory (ZFC) cannot settle it — either way.
Just as many points
Laboratorium · Zeno and series
Achilles and the tortoise
Zeno of Elea (c. 450 BC): before Achilles catches the tortoise, he must first reach the spot where the tortoise was. But by then the tortoise has moved on — and so on forever. So he never catches it?
We show each step at the same pace. That's an illusion: in real time the steps keep getting shorter.
A square that fills up
The sum of a convergent series is the limit of its partial sums. Geometric series: a + aq + aq² + ⋯ = a / (1 − q) for |q| < 1.
0.999… = 1. Really.
1. Thirds
13 = 0.333…, so 3 · 13 = 0.999… — and 3 · 13 is 1.
2. Shifting the decimal point
x = 0.999…, 10x = 9.999…, so 10x − x = 9, that is 9x = 9 and x = 1.
3. Nothing in between
Between two different real numbers there is always another one, e.g. their average. Try to find a number greater than 0.999… but less than 1:
“0.999…” is not a number that “keeps growing” — it is the limit of the sequence 0.9, 0.99, 0.999, … And that limit is exactly 1. One number simply has two decimal expansions.
A tower over the abyss
The harmonic series 1 + ½ + ⅓ + ¼ + ⋯ grows without bound. That is why identical blocks can be stacked into a tower whose top block sticks out beyond the table as far as you like: with n blocks, by at most Hn/2 block lengths.
The tower stands if the center of mass of the blocks above each layer lies over the block below.
Model: rigid, identical blocks, one per layer. With 100 blocks the overhang is about 2.59 lengths; with a million, about 7.2.
The harmonic series: it grows, but how slowly!
Nicole Oresme (c. 1350): ½ + (⅓ + ¼) + (⅕ + … + ⅛) + ⋯ — each group gives more than ½, and there are infinitely many groups. Approximation: Hn ≈ ln n + γ, γ ≈ 0.5772.
Grandi's series: 1 − 1 + 1 − 1 + ⋯
(1 − 1) + (1 − 1) + ⋯ = 0, but 1 + (−1 + 1) + (−1 + 1) + ⋯ = 1. Which result is right?
Rearrange the series, change the sum
1 − ½ + ⅓ − ¼ + ⋯ = ln 2 ≈ 0.693. But the positive and the negative terms on their own each add up to infinity. Riemann's theorem: by reordering, you can get any sum you like. Recipe: add positive terms until you pass the target; then subtract negative ones until you drop below it; and repeat.
Laboratorium · fractals and infinite shapes
Infinite perimeter, finite area
Dimension of a segment: 1
Shrink a segment 3 times — you need 3 copies to rebuild it. 3 = 31.
Dimension of a square: 2
Shrink a square 3 times — you need 9 copies. 9 = 32. In general: N = sD, so D = log N / log s.
Dimension of the Koch curve: 1.26
Shrink it 3 times — you need 4 copies. D = log 4 / log 3 ≈ 1.262. Sierpiński triangle: 3 copies at scale ½, D = log 3 / log 2 ≈ 1.585. This is the fractal dimension: something between a line and a surface.
The coastline paradox
Lewis Fry Richardson measured borders and coastlines with dividers. He noticed that the shorter the “ruler,” the longer the result. A coastline has no single length — what matters is how fast the length grows.
On a log–log scale the measurements lie on a line with slope 1 − D. Richardson's result for the west coast of Great Britain: D ≈ 1.25; Norway's fjord-ragged coast has an even higher dimension.
Gabriel's horn (Torricelli's trumpet)
Evangelista Torricelli discovered it in the 1640s and astonished the scholars of his day: a solid of infinite length, yet with finite volume.
Laboratorium · paradoxes
The Banach–Tarski paradox (1924)
Two Polish mathematicians, Stefan Banach and Alfred Tarski, proved that a solid ball can be cut into finitely many pieces which — only rotated and moved — can be reassembled into two balls identical to the original.
- Ball. An ordinary solid ball: uncountably many points.
- Five “pieces”. Not solids, but scattered clouds of points with no defined volume.
- Rotations and translations. No stretching or squeezing.
- Two balls. Each identical to the first.
The infinite monkey
29 keys: 26 letters, space, comma, periodA monkey hits the keys of a typewriter at random. If it types forever, then with probability 1 it will type any text. But how long will that take?
The formula accounts for the text overlapping itself: “TO BE” is both at the start and at the end, so the average is 2918 + 295. Age of the Universe: about 1.38 · 1010 years. More about randomness: Probability.
Thomson's lamp (1954)
The lamp is off. After ½ minute we switch it on, after another ¼ we switch it off, after ⅛ on again… After one minute we have flipped the switch infinitely many times. Is the lamp on?
The barber paradox (Russell, 1901)
The barber shaves all the men in town who do not shave themselves — and only them. Does the barber shave himself?
The set of all sets that are not members of themselves
Bertrand Russell's version for mathematicians: let R = {x : x ∉ x}. If R ∈ R, then by definition R ∉ R — and vice versa. That is a real contradiction, not just a surprise.
Gödel: truths that can't be proved (1931)
Kurt Gödel showed that in every consistent system of axioms that covers the arithmetic of the natural numbers (and whose axioms can be listed by a recipe), there are statements that can be neither proved nor disproved. The infinite world of numbers is richer than any finite set of rules.
Laboratorium · infinity in the world
Is anything in nature infinite?
Everything we have ever measured fits into about 62 orders of magnitude: from the Planck length (1.6 · 10−35 m) to the diameter of the observable Universe (about 93 billion light-years, 8.8 · 1026 m). Slide the ruler.
More: Scale of the Universe, Elementary particles, Black hole.
Is the Universe infinite?
Nobody knows. We see only part of it — the observable Universe, which is finite because light has had only a finite time to travel. Measurements show that space is nearly flat, which fits both a very large and an infinite Universe.
Can matter be divided forever?
Not the way the ancients thought. Matter is made of atoms (about 10−10 m), atoms of nuclei and electrons, and nuclei of quarks. The Planck length is the scale at which current theories stop working — not a proven “pixel” of space.
Infinity as an alarm signal
When a physics formula gives ∞ — like the density at the center of a black hole — physicists usually read it as a sign that the theory has gone beyond its range, not as a truly infinite value.
Computers and infinity
Numbers in JavaScript (the IEEE 754 format) have a special value, Infinity. Click — your browser computes the result:
Keep doubling until it overflows
Where does the ∞ sign come from?
The symbol ∞ was introduced by the English mathematician John Wallis in 1655 in a treatise on conic sections. Where the shape comes from is not known for sure; one hypothesis links it to a Roman way of writing the number 1000.
In mathematics, ∞ is not a number you can calculate with like any other. “x → ∞” is shorthand: x grows beyond every bound. With the cardinalities of sets (ℵ0, 𝔠), however, we are talking about specific, different infinities.
A brief history of infinity
gold frames: the Polish thread- c. 450 BCZeno of Elea — the paradoxes of motion: Achilles and the tortoise, the arrow, the dichotomy.
- 3rd c. BCArchimedes — the method of exhaustion; the area of a parabolic segment is 4/3 of a triangle, i.e. the sum 1 + ¼ + ¹⁄₁₆ + ⋯
- c. 1350Nicole Oresme — proof that the harmonic series diverges.
- 1638Galileo — in his book on “two new sciences” (Discorsi), he notes that there are as many squares as natural numbers.
- 1655John Wallis — introduces the symbol ∞.
- 1874 and 1891Georg Cantor — the real numbers cannot be numbered; later the diagonal method and the power set theorem.
- 1904Helge von Koch — a curve with no tangent at any point: the Koch snowflake.
- 1915–1916Wacław Sierpiński (1882–1969) — the Sierpiński triangle and carpet; co-founder of the Warsaw School of Mathematics and the journal Fundamenta Mathematicae (1920), a great researcher of set theory.
- 1924Stefan Banach and Alfred Tarski — the ball-doubling paradox, published in Fundamenta Mathematicae.
- 1924–1925David Hilbert — in his lectures “On the Infinite,” he tells of a hotel with infinitely many rooms; George Gamow (1947) made it widely known.
- 1931 and 1938–1940Kurt Gödel — the incompleteness theorems; later a proof that the continuum hypothesis cannot be disproved in ZFC.
- 1963Paul Cohen — uses the method of forcing to show that the continuum hypothesis cannot be proved either.
MathematicsHow to count infinity
Cardinality and bijection
Two sets have the same cardinality if there is a bijection between them — a one-to-one correspondence in which every element has exactly one partner. A set is infinite exactly when it is the same size as some proper part of itself (Dedekind's definition). Hilbert's Hotel and Galileo's paradox show exactly this.
The last formula is the diagonal zigzag: passenger k from bus b (b = 0 means the original guests) lies on diagonal d = b + k − 1, at position j (counted alternately from the top and from the bottom). Everyone gets a single room of their own, and every room gets a guest.
Countable and uncountable
The natural numbers, the integers and the rationals can be numbered — they are countable sets. The real numbers cannot: Cantor's diagonal turns any list into a number not on it. A segment, a line and a square all have the same number of points — the cardinality of the continuum.
Series: convergent and divergent
The sum of a series is the limit of its partial sums, if it exists — then the series is convergent. Achilles takes infinitely many steps, but their durations form a convergent series: 10 + 1 + 0.1 + ⋯ = 11.11… s. The harmonic series is divergent, even though its terms shrink to zero — it passes 10 only after 12,367 terms. The alternating harmonic series 1 − ½ + ⅓ − ⋯ = ln 2 converges only conditionally, so rearranging its terms changes its sum.
Fractals
A fractal is a shape whose parts, when magnified, resemble the whole. Its fractal dimension D tells you how many (N) copies scaled down s times you need to rebuild the whole. Gabriel's horn combines a finite volume π with an infinite surface area.
Paradox or contradiction?
Most “paradoxes of infinity” are surprises: they defy intuition, but not logic (the hotel, Galileo, Zeno, Gabriel's horn, Banach–Tarski). Russell's paradox is a genuine contradiction — it was removed by building set theory on axioms. Thomson's lamp and Grandi's series show that not every infinite process has an outcome.
Simplifications on this page
The screen always shows a finite fragment: a few rooms, digits, list rows, steps or fractal levels. The coastline is computer-generated and has limited precision, so with a very short ruler the result would stop growing. The Banach–Tarski animation is only a metaphor — the real pieces can't be drawn. The tower of blocks assumes rigid, identical blocks and exact positions.
Did you know…?
- There are exactly as many even numbers as natural numbers: just pair them up, n ↔ 2n. The same goes for the primes and for fractions.
- Wacław Sierpiński (1882–1969), creator of the famous triangle, was one of the pillars of the Warsaw School of Mathematics. He wrote more than 700 papers, and a crater on the Moon is named after him.
- Gabriel's horn has a finite volume π but an infinite surface area: there is enough paint to fill it, but not to coat its walls with a layer of constant thickness.
Challenges
- Infinite busCheck an infinite busload into the full hotel.not completed
- Not on the listBuild a number that is not on the list using the diagonal method.not completed
- Twenty Zeno stepsTake Achilles through 20 Zeno steps.not completed
- Rearranged seriesRearrange the series so that its sum is 2.not completed
- Koch snowflakeDraw a level-5 Koch snowflake and compare the perimeter.not completed
- Block over the abyssBy dragging the blocks, make a block stick out beyond the table edge by more than its own length.not completed
Challenges check themselves. Your progress is saved in this browser.
See also
Euclid proved that there are infinitely many primes; π can be written as an infinite series; and in “Limits and integrals” you can compute the area under 1/x² just as for Gabriel's horn.