Here are some very ill-thought-out ideas on Kolmogorov complexity.

We define a metric on the space of bit-strings . For a universal Turing machine , let be the “length” of the shortest program that outputs on input , or outputs on input . This should measure how difficult it is to “relate” and .

The ends of the metric space should correspond to infinite random bitstrings, and because choosing a different univeral Turing machine just replaces this metric space with one quasi-isometric to it, the ends should be preserved, so there will still be a lot of infinite random bitstrings

But obviously I haven’t thought about any of this very carefully: for instance, the triangle inequality probably only holds coarsely, because it depends on being able to concatenate programs.

Here’s a similar question. Usually, we start with a partial function which tells how to translate descriptions into objects; Kolmogorov complexity is then defined as . Any universal Turing machine gives a measure of complexity with the same asymptotics, i.e., and differ by a constant that depends only on and . Suppose I have another function so that has the same asymptotics: what more can be said about ?

There’s a stupid rigidity for computable functions (a computable function is still computable if its value is changed at finitely many places), and maybe these sort of questions could lead to a rigidity theorem for computability, a local Church-Turing thesis.

And having written this, I’m terrified at how similar I sound to Archimedes Plutonium. Now I’ll go to learn more about localization of spaces in the algebraic topology proseminar.