Unprovability, phase transitions and the Riemann zeta-function

Unprovability Theory started with Kurt Gödel’s incompleteness theorems in 1931 but only gained mathematical significance since the late 1970s when Jeff Paris and Harvey Friedman discovered the first few families of interesting combinatorial statements that cannot be proved using the axioms of Peano Arithmetic or even some stronger axiomatic systems. In this survey article we briefly introduce the subject of Unprovability Theory to non-logicians and describe two of its directions that have recently been pursued by the authors, namely phase transitions and encodings of Ramsey-like statements using the Riemann zeta-function. Phase transitions between provability and unprovability of parameterised families of assertions were introduced by the second author in 2000. We give several examples and sketch some explanations of the reasons behind this phenomenon. Unprovability results that involve the Riemann zeta function are consequences of classical results about the Riemann zeta-function: Bohr’s almost periodicity and Voronin’s universality theorems. We also indicate how the universality phenomenon will give us many more unprovable assertions in the future.