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Harvey Friedman: My forty years on his shoulders
Abstract
I 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.
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