A major theme in proof theory consists in classifying the proof strength of mathematical frameworks for reasoning about mathematics. The resulting phase transitions from provability to unprovability are interesting from the foundational as well as from the mathematical point of view. It is very surprising that during these investigations methods from analytic number theory, combinatorial probability theory, complex analysis and Ramsey theory enter the scene.
In this abstract we present the underlying research program. We try to explain our intuition about the nature of phase transition results under discussion. It is our objective to show that there are intriguing interrelations between “usual” mathematics and proof theory.
The abstract is intended to be accessible to a general mathematical audience. We therefore concentrate on basic examples and ideas without giving proofs.

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