Inductive Complexity Measures for Mathematical Problems

An algorithmic uniform method to measure the complexity of statements of finitely refutable statements [6, 7, 8], was used to classify fa-mous/interesting mathematical statements like Fermat's last theorem, Hilbert's tenth problem, the four colour theorem, the Riemann hypothesis, [9, 13, 14]. Working with inductive Turing machines of various orders [1] instead of classical computations, we propose a class of inductive complexity measures and inductive complexity classes for mathematical statements which generalise the previous method. In particular, the new method is capable to classify statements of the form 8n9m R(n, m), where R(n, m) is a computable binary predicate. As illustrations, we evaluate the inductive complexity of the Collatz conjecture or twin prime conjecture—which cannot not be evaluated with the original method.