On Characteristic Constants of Theories Defined by Kolmogorov Complexity

Chaitin discovered that for each formal system T , there exists a constant csuch that no sentence of the form K(x) > cis provable in T , where K(x) is the Kolmogorov complexity of x. We call the minimum such cthe Chaitin characteristic constant of T , or c T . There have been discussions about whether it represents the information content or strength of T . Raatikainen tried to reveal the true source of c T , stating that it is determined by the smallest index of Turing machine which does not halt but we cannot prove this fact in T . We call the index the Raatikainen characteristic constant of T , denoted by r T . We show that r T does not necessarily coincide with c T ; for two arithmetical theories T , Ti¾? with a i¾? 1 -sentence provable in Ti¾? but not in T , there is an enumeration of the Turing machines such that r T Ti¾? and c T = c Ti¾? .