Proof interpretations: theoretical and practical aspects

We study theoretical and practical aspects of proof theoretic tools called proof interpretations. (1) Theoretical contributions. (1.1) Completeness and omega-rule. Using a proof interpretation, we prove that Peano arithmetic with the omega-rule is a complete theory. (1.2) Proof interpretations with truth. Proof interpretations without truth give information about the interpreted formula, not the original formula. We give three heuristics on hardwiring truth and apply them to several proof interpretations. (1.3) Copies of classical logic in intuitionistic logic. The usual proof interpretations embedding classical logic in intuitionistic logic give the same copy of classical logic, suggesting uniqueness. We present three different copies. (2) Practical contributions. (2.1) "Finitary" infinite pigeonhole principles. Terence Tao studied finitisations of statements in analysis. We take a logic view at Tao's finitisations through the lenses of proof interpretations and reverse mathematics. (2.2) Proof mining Hillam's theorem. Hillam's theorem characterises the convergence of fixed point iterations. We proof mine it, getting a "finitary rate of convergence" of the fixed point iteration.