Slicing the Truth
2014 book by Denis Hirschfeldt
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to prove combinatorial theorems. It was written by Denis R. Hirschfeldt, based on a course given by Hirschfeldt at the National University of Singapore in 2010, and published in 2014 by World Scientific, as volume 28 of the Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore.
Topics
The book begins with five chapters that discuss the field of reverse mathematics, which has the goal of classifying mathematical theorems by the axiom schemes needed to prove them, and the big five subsystems of second-order arithmetic into which many theorems of mathematics have been classified. These chapters also review some of the tools needed in this study, including computability theory, forcing, and the low basis theorem.
Chapter six, "the real heart of the book", applies this method to an infinitary form of Ramsey's theorem: every edge coloring of a countably infinite complete graph or complete uniform hypergraph, using finitely many colors, contains a monochromatic infinite induced subgraph. The standard proof of this theorem uses the arithmetical comprehension axiom, falling into one of the big five subsystems, ACA0. However, as David Seetapun originally proved, the version of the theorem for graphs is weaker than ACA0, and it turns out to be inequivalent to any one of the big five subsystems. The version for uniform hypergraphs of fixed order greater than two is equivalent to ACA0, and the version of the theorem stated for all numbers of colors and all orders of hypergraphs simultaneously is stronger than ACA0.
Chapter seven discusses conservative extensions of theories, in which the statements of a powerful theory (such as one of the forms of second-order arithmetic) that are both provable in that theory and expressible in a weaker theory (such as Peano arithmetic) are only the ones that are already provably in the weaker theory. Chapter eight summarizes the results so far in diagrammatic form.
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