The axiomatic power of Kolmogorov complexity

TL;DR

This paper explores the power and limitations of adding Kolmogorov complexity-based axioms to formal theories, showing they can prove all true -statements but are limited in deriving new theorems, especially when considering proof sizes.

Contribution

It provides a detailed analysis of the axiomatic strength of Kolmogorov complexity statements and compares different formalizations of randomness within logical systems.

Findings

Adding all true -statements of Kolmogorov complexity yields a theory proving all true -statements.

Random axioms do not significantly extend provable theorems unless proof size considerations are included.

Different formalizations of Martin-Lf6f randomness are closely related but not equivalent.

Abstract

The famous G\"odel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates 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 axioms 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 answers a question recently asked by Lipton. 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. We then study the axiomatic power of the statements of type "the Kolmogorov complexity of x exceeds n" in general. They are \Pi_1 (universally quantified) statements of Peano arithmetic. We show that by adding all true statements of this type, we obtain a theory that proves all true \Pi_1-statements, and also provide a more detailed classification. In particular, to derive all true \Pi_1-statements it is sufficient 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. We also study other logical questions related to "random axioms". Finally, we consider a theory that claims Martin-L\"of 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.