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
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