lets pretend “number of bits needed to describe X” means “log2 the length of the shortest theory that proves the existence of X.”
Whether a theory proves the existence of X may be an undecideable question.
How many bits it takes to describe X is an undecidable question when defined in other ways, too.
The definition “length of the shortest program which minimizes (program length + runtime)” isn’t undecideable, although you could argue that that’s not what we normally mean by number of bits.
Adding program length and runtime feels to me like a type error.
Whether a theory proves the existence of X may be an undecideable question.
How many bits it takes to describe X is an undecidable question when defined in other ways, too.
The definition “length of the shortest program which minimizes (program length + runtime)” isn’t undecideable, although you could argue that that’s not what we normally mean by number of bits.
Adding program length and runtime feels to me like a type error.