Transcendental Proofs

Harvey M. Friedman (OSU) gives a talk at the MCMP Colloquium (30 October, 2012) titled "Transcendental Proofs". Abstract: We discuss our Transcendental Proofs project, spanning approximately 45 years, which aims to uncover mathematically fundamental, rich, and diverse subareas of athematics which can only be developed by going well beyond the usual ZFC axioms for mathematics. The discussion will have mathematical, computational, and philosophical components.The mathematical component focuses on long finite sequences, adjacent amsey theorems, finite trees and graphs, continuous maps between countable sets, Borel diagonalization and selection, Boolean Relation Theory, and Embedded Maximal Cliques. The computational component focuses on predictions from higher infinities of the result of actual computations. The philosophical component focuses on various formulations of the project, involving measures of simplicity and naturalness, as well as prospects for confirmation of the validity of transcendental methods.