In Solomonoff Induction, which purports to be a theory of universal inductive inference, the "complete hypothesis space" consists of all computable programs (note that all current physical theories are computable, so this hypothesis space is very general). Then induction is performed by keeping all programs consistent with the observations, weighted by 2 terms: the programs prior likelihood, and the probability that program assigns to the observations (the programs can be deterministic and assign probability 1).
The "prior likelihood" in Solomonoff Induction is the program's complexity (well, 2^(-Complexity), where the complexity is the length of the shortest representation of that program.
Altogether, the procedure looks like: maintain a belief which is a mixture of all programs consistent with the observations, weighted by their complexity and the likelihood they assign to the data. Of course, this procedure is still limited by the sample/observation space!
That's our best formal theory of induction in a nutshell.