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Surprises in Logic (2016)

There's a complexity barrier built into the very laws of logic: roughly speaking, while lots of things are more complex than this, we can't prove any specific thing is more complex than this. And this barrier is surprisingly low! Just how low? Read this.

Such restrictions provide a challenge for coders, musicians and graphics artists and bring back the old motive of making a device do more than was intended in its original design. The complete genetic code of an organism can be as short as a few hundred thousand bytes, and that has to be encoded in a way that doesn't allow for highly clever compression schemes. $$ U + \log_2(L) + K \lt L $$\(U\) is the length of a program where if you input a natural number \(i\), it will search through all proofs in Peano arithmetic until it finds one that proves some bit string has Kolmogorov complexity \( > i\).

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