Is Kolmogorov-random nonsensical for small numbers? The Wikipedia definition of Kolmogorov-random defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. 

Aren't nearly1 all strings shorter than a certain length2 going to be Kolmogorov-random?


If a random Turing machine is chosen, then programs that output short strings will almost always be longer than the strings. If the programming language used for measurement is optimized for outputting a short string, then only the complexity of a subset of short strings can be made lower than their length. For example, if a Turing complete language defines $0$ as an operation that outputs 0 and halts, the program for outputting 1 and halting is forced to be longer than the length of its output. 
The exact length below which this occurs will depend on the programming language chosen. The length is 41 bits using 5 state, 2 symbol Turing machines.
Is Kolmogorov-random nonsensical for small numbers?



Although one could pick a specific description language to make Kolmogorov complexity into

a function from ​ {0,1}* ​ to ​ {0,1,2,3,...} , ​ most people don't pick as described ​ ( 0 , 1 , 2 , 3 ) ,

so basically:

Kolmogorov complexity is only defined up to an additive O(1), so one

needs an infinite set of strings to make sense of Kolmogorov randomness.

In particular, yes, Kolmogorov-randomness is nonsensical for small numbers.