The axiomatic power of Kolmogorov complexity

The famous Godel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true state- ments that are unprovable in T. Such statements would be natural can- didates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T. In this paper we discuss another approach motivated by Chaitin’s version of G odel’s theorem where ax- ioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us to prove new interesting theorems. This result (cf. [She06]) answers a question recently asked by Lipton [LR11]. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE). This result partially answers the question asked in [She06]. We then study the axiomatic power of the statements of type “the Kolmogorov complexity of x exceeds n” (where x is some string, and n is some integer) in general. They are Π1 (universally quantified) statements of Peano arithmetic. We show (Theorem 5) that by adding all true statements of this type, we obtain a theory that proves all true Π1-statements, and also provide a more detailed classification. In particular, as Theorem 7 shows, to derive all true Π1-statements it is enough to add one statement of this type for each n (or even for infinitely many n) if strings are chosen in a special way. On the other hand, one may add statements of this type for most x of length n (for every n) and still obtain a weak theory (Theorem 10). We also study other logical questions related to “random axioms” (hierarchy with respect to n, Theorem 8 in Section 3.3, independence in Section 3.6, etc.). Finally, we consider a theory that claims Martin-Lof randomness of a given infinite binary sequence. This claim can be formalized in different ways. We show that different formalizations are closely related but not equivalent, and study their properties.