Harvey Friedman: My forty years on his shoulders
AbstractI will discuss my efforts to relate the great insights of Kurt Gödel directly to ordinary mathematical practice. We will emphasize that this is an evolutionary process which is only at its very initial stages of development. The initial focus will be on the necessary use of impredicative methods for the proofs of the celebrated theorems of J.B. Kruskal and Robertson/Seymour in finite trees and finite graphs. We will later focus on the necessity of going beyond all of the usual accepted axioms of mathematics (ZFC) in the contexts of Borel measurable functions (Borel selection), infinite sets of natural numbers (Boolean Relation Theory), and finite graph theory (finite digraphs on lattice points). A critical ingredient throughout this development is the epochal Gödel Second Incompleteness Theorem.