The Tegmark IV multiverse is so big that a human brain can’t comprehend nearly any of it, but the theory as a whole can be written with four words: “All mathematical structures exist”. In terms of Kolmogorov complexity, it doesn’t get much simpler than those four words.
To be fair, every one of those words is hiding a substantial amount of complexity. Not as much hidden complexity as “A wizard did it” (even shorter!), but still.
Actually, I’m quite unclear about what the statement “All mathematical structures exist” could mean, so I have a hard time evaluating its Kolmogorov complexity. I mean, what does it mean to say that a mathematical structure exists, over and above the assertion that the mathematical structure was, in some sense, available for its existence to be considered in the first place?
ETA: When I try to think about how I would fully flesh out the hypothesis that “All mathematical structures exist”, all I can imagine is that you would have the source code for program that recursively generates all mathematical structures, together with the source code of a second program that applies the tag “exists” to all the outputs of the first program.
Two immediate problems:
(1) To say that we can recursively generate all mathematical structures is to say that the collection of all mathematical structures is denumerable. Maintaining this position runs into complications, to say the least.
(2) More to the point that I was making above, nothing significant really follows from applying the tag “exists” to things. You would have functionally the same overall program if you applied the tag “is blue” to all the outputs of the first program instead. You aren’t really saying anything just by applying arbitrary tags to things. But what else are you going to do?
Actually, I’m quite unclear about what the statement “All mathematical structures exist” could mean, so I have a hard time evaluating its Kolmogorov complexity. I mean, what does it mean to say that a mathematical structure exists, over and above the assertion that the mathematical structure was, in some sense, available for its existence to be considered in the first place?
ETA: When I try to think about how I would fully flesh out the hypothesis that “All mathematical structures exist”, all I can imagine is that you would have the source code for program that recursively generates all mathematical structures, together with the source code of a second program that applies the tag “exists” to all the outputs of the first program.
Two immediate problems:
(1) To say that we can recursively generate all mathematical structures is to say that the collection of all mathematical structures is denumerable. Maintaining this position runs into complications, to say the least.
(2) More to the point that I was making above, nothing significant really follows from applying the tag “exists” to things. You would have functionally the same overall program if you applied the tag “is blue” to all the outputs of the first program instead. You aren’t really saying anything just by applying arbitrary tags to things. But what else are you going to do?