Let L ag ¼ fþ; À; 0g be the language of the abelian groups, L an expansion of L ag ð<Þ by relations and constants, and L mod ¼ L ag U f1 n g nb2 where each 1 n is defined as follows: x 1 n y if and only if n j x À y. Let H be a structure for L such that H j L ag ð<Þ is a totally ordered abelian group and K a totally ordered abelian group. We consider a product interpretation of H  K with a new predicate I for f0g  K defined by N. Suzuki [9].

No 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.