A list of ramblings that could prove useful in extending Solomonoff Induction, or that could well be all false:

• Kolmogorov complexity is a way to assign to each explanation a natural number

• assigning a natural number to a program is a way to pidgeon-hole the totality of programs to a well ordered countable set, in such a way that no pidgeon-hole has infinite pidgeons in it

• if every partion of k^n in m parts has an homogeneous set of size j, then k --> j^n_m

• let w be omega, n, m finite, then w --> w^n_m (Ramsey theorem)

• w -/-> w^n_w, on this you can construct Solomonoff induction: partition the set of explanation in such a way that no group of explanation is infinite, then every hypothesis has measure 1/k2^-n, where k is the cardinality of the group and n is the position of the group's pidgeon-hole. No notion of complexity is needed, although complexity is a way to partition explanations

• but also 2^k -/-> w^2_k, for every cardinal k.

• This means 2^w -/-> w^2_w, so you could make SI work on infinite explanations too

• extend to every k --> j^n_m so that j and m are measurable