Assume HALT exists. Define:
DIAGONAL(x):
if HALT(x, x) loops forever
return "halted"
else
loop forever ← the trap
Now ask: does DIAGONAL(DIAGONAL) halt?
HALT cannot exist. Not due to engineering limits — due to logic.
The impossibility is woven into the definition of computation itself.
For any integer n:
try: 27
steps until 1: —
Every number tested eventually reaches 1. No proof it always must.
Tested to 268. True for all. Proven for none.
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Church encoded numbers as pure functions:
0 = λf.λx. x 1 = λf.λx. f x 2 = λf.λx. f (f x) 3 = λf.λx. f (f (f x))
Try:
Every number is a function that applies f n times to x.
Addition = function composition. Multiplication = composition of compositions.
All of computation from nothing but anonymous functions.
"This statement has no proof in arithmetic."
If G is provable → G is false → arithmetic is inconsistent.
If G is true → arithmetic is consistent but incomplete.
● truth ● provable ● true, not provable
Kurt Gödel, 1931. He encoded this as a number.
A number that speaks about its own provability.
A sphere can be decomposed and reassembled into two spheres of the same size.
Uses the axiom of choice. The pieces are non-measurable —
no defined volume. The paradox lives at the foundations of set theory.
A theorem, not a conjecture. It has been proven.
Hover over the sphere to see the decomposition.
Four rules. Infinite complexity.
Any live cell with 2–3 live neighbors survives.
Any dead cell with exactly 3 live neighbors becomes alive.
All other cells die or stay dead.
Click on the grid to toggle cells. Press play to watch
patterns emerge from randomness — gliders, oscillators, stable blocks.
John Conway, 1970. Computationally universal.
BB(n) = the maximum steps any n-state Turing machine can run before halting.
BB(5) = 47,176,870. Proven in 2006 by Allison, Heule, Kott
BB(6) remains unknown — but we know it's finite and larger than BB(5).
And BB(7) > BB(6). And BB(8) > BB(7). And so on, forever.
BB grows faster than any computable function.
If BB were computable, you could solve the halting problem. Contradiction.
Therefore BB is uncomputable — and yet we know some of its values.