Rikai Funo · Track 16 · middle
Ode to Algorithmic Information Theory
The complexity of a string is the length of the shortest program that generates it. Most strings are incompressible — their shortest description is themselves, random all the way through. Chaitin's Omega is the probability that a randomly constructed program halts: real-valued, well-defined, and utterly uncomputable — a specific number that encodes all of mathematical truth. An ode to complexity as the inability to compress, and the number that contains every theorem.
Lyrics
There is a number. Just one. A secret written on a single page. Kolmogorov said it first in 1963. Most things, you can't shorten them. Their only true name is themselves. This string of characters, this life, this moment. No summary will do. Incompressible. Random all the way through. Yorktown Heights, 1975. Gregory Chaitin at the IBM 360. The air is cool, 68 degrees. He asks the machine a simple question. What is the chance a random program stops? And the line printer hums back an answer. A real number. Oh, Omega. My beautiful, impossible Omega. The first sixty-four bits fit on a single sheet of paper. The probability of a halt. A shadow cast by all of mathematics. A number we can name but never know. And hidden in your digits, past the part we can see... Is the answer to everything. Every theorem, every conjecture. True or false. Encoded in the endless binary dance. The quiet, perfect library of what is. If only we could read that far. Each new bit costs more than the last. An eternity of computation for the next zero. The next one. A wall of pure logic. We stand at the beginning of you. Holding the first page. Knowing the book is infinite. The first sixty-four bits. A whisper from the machine in 1975. The rest is silence. The rest is waiting.