Kolmogorov complexity and characteristic constants of formal theories of arithmetic

Abstract

MSC (2010) 03A99, 03B25We investigate two constants c T and r T , introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that c T does not represent the complexity of T and found that for two theories S and T, one can always find a universal Turing machine such that c S = c T . We prove the following are equivalent: c S = c T for some universal Turing machine, r S = r T for some universal Turing machine, and T proves some Π1 -sentence which S cannnot prove.