Okay, but 2 points: 1) Kolmogorov complexity is a measure of the computational resources needed to specify the object Depending on what you mean by "specify t

Okay, but 2 points: 1) Kolmogorov complexity is a measure of the computational resources needed to specify the object Depending on what you mean by "specify the object" this may or may not be infinite resources. My vote goes to infinite. Because you either need infinite resources even to just express this number, or you need infinite resources to actually run it. I get that you can create a finite program that gives an approximation of the number, but that's not the same, and only approximations can be finite. The number itself, and it's useful specifications, are all infinite. For the real number you need infinite resources to express it. So by what measure exactly does sqrt(2) have finite complexity ? In fact, you could use the old "every number occurs in Pi" trick and say that every other object is contained within it (and within every other real number), and since kolmogorov complexity is a metric I have just proven that the kolmogorov complexity of sqrt(2) is larger than any other kolmogorov complexity. The only number that satisfies that condition is infinite. 2) Even if this applies to the 4 numbers I specified, or the few thousand "well known" real numbers, it doesn't apply to any real proportion of real numbers. I mean, I can compute one or two 3rd degree boundary differential equations straight in my head, but that is very different from saying I can do that in the general case. Hell, quantum computers can't do it in the general case. The same is true for specifying real numbers. (and, more or less) 3) in reality, only approximations of sqrt(2) are computable. The real number itself isn't.