PARADOX.com
an instrument catalog of impossible things
Coherence100.0%
stable
Vol. I — six exhibits, all real mathematics
Each plate below is a working demonstration, not an illustration — drag the sliders, click the planks, ask the barber the question. Nothing here is a trick of language. The numbers really do behave like this.
Plate 0 — the flagship
Torricelli's Trumpet, 1641 — rotate y = 1/x around the x-axis from 1 to L
Spin the curve y = 1/x around the x-axis and let it run out to infinity. The volume of the resulting horn converges — it never exceeds π cubic units, no matter how far you extend it. The surface area, meanwhile, grows without bound. Pour paint in and it fills completely, using a finite, fixed amount. That same paint, spread as a coating, would have to cover an infinite surface. The horn holds exactly enough paint to fill itself, and never enough to paint itself.
Plate I
Elea, c. 450 BCE — to reach the wall, first cross half the distance
Before you can travel the full distance to the wall, you must first travel half of it. Then half of what remains. Then half of that. Every step is real progress, and the wall is never reached — because there is always another half left.
Plate II
David Hilbert, 1924 — a hotel with infinitely many rooms, every one occupied
The hotel is full. Infinitely full — every one of its infinite rooms has a guest. And yet it can always take one more: move the guest in room n to room n+1, for every room, all at once. Room 1 is now free. Nobody was evicted. The hotel was full both before and after.
every room occupied — the hotel is, as ever, full.
Plate III
Plutarch, c. 75 CE — replace every plank, one at a time
The ship sails for years. Each time a plank rots, the crew swaps it for new timber. Eventually, not one original plank remains — yet at no single moment did the ship stop being itself.
original planks remaining: 100%
Zero original planks remain. Is it still the same ship?
Plate IV
Bertrand Russell, 1901 — a rule that cannot describe its own maker
In a certain village, the barber shaves every man who does not shave himself — and only those men. A simple, well-formed rule. Now: who shaves the barber?
Does the barber shave himself?
Plate V
popular in time-travel fiction since the 1930s — a loop with no first cause
You travel back before your grandparents met, and prevent the meeting. If they never meet, you are never born. If you are never born, you cannot travel back to prevent it. If you cannot prevent it, they meet after all — and you are born, and travel back, and prevent it.