In its simplest form the famous first Gödel incompleteness theorem states that there are true statements about natural numbers which do not follow from the standard axiom system to reason about natural numbers. Quite a few times such statements undergo a phase transition. An example for a phase transition in real life is given by the transition of matter from ice to water or from water to steam. For a phase transition in logic imagine that an assertion A which depends on a parameter function f such that A is provable for slow growing f and such that A (still being true) becomes unprovable for fast growing f. In this project we will explore the thresholds for the transition from provability to unprovability. To obtain good characterizations of the phase transition we combine logical methods with methods from other areas in mathematics like real analysis and combinatorics. In particular we will employ rather advanced methods from the theory of ordinals and the theory of subrecursive hierarchies.
We will classify the phase transition for Goodstein principles. As spin offs we obtain results regarding the maximal lengths of Goodstein sequences, modified Goodstein principles, and functorial Goodstein theorems. Moreover we will classify the phase transition for Friedman's  Bolzano-Weierstrass theorem. We conjecture that as a spin off we will obtain unexpected and far reaching applications of Ecalle's transseries to proof theory.