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Five papers on reverse mathematics and Ramsey-theoretic principles

motivated by the hope that a better understanding and new techniques for dealing with weaker systems might make a return to the analysis of stronger systems more viable. The other technical papers in the volume are still based on Gentzen’s techniques but apply them to other areas of proof theory and logic. In the realm of pure proof theory,Meskens and Weiermann investigate fast-growing functions in subsystems of Peano arithmetic, Oliva and Powell give a game-theoretic version of Gödel’s Dialectica interpretation, Ferreira discusses Spector’s consistency proof for analysis using higher order functionals, and Jervall discusses tree representations of ordinals. Further from proof theory, Rathjen and Vizcaı́no look at the connection to reverse mathematics, giving an equivalence between a reverse mathematics system and the statement that a certain operation inspired by proof-theoretic ordinals maps well-orderings to well-orderings, and Pohlers discusses so-called semiformal calculi (proof systems where rules may have infinitely many premises), with a focus on applications outside of proof theory. The remaining papers in the volume consider the broader impact of Gentzen’s ideas. Kahle’s contribution gives a survey of consistency results and how they have been viewed, both historically since Gentzen’s work and across several subfields of modern logic. Detlefsen discusses Gentzen’s own philosophical views, and Setzer considers the revision of Hilbert’s original program to prove the consistency of mathematics in the context of modern proof theory. BecauseGentzen’s cut-elimination results for Peano arithmetic are so central to the volume, even the less technical papers will be difficult to fully appreciate for readers who are not comfortable with that proof. However it is difficult to imagine a more thorough investigation of what that result means eighty years after its publication, both in the broad scope of its consequences, the field it generated, and the way it transformed our understanding of what consistency means in mathematics.