Abstract:
An algorithmic uniform method to measure the complexity of
statements of finitely refutable statements [6, 7, 8], was used to classify famous/
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 ∀n∃m 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.