Speaker Series: James Walsh (NYU)

Title: The Boundaries of Incompleteness

Abstract: Are the fundamental principles of mathematical reasoning consistent? An early goal of mathematical logic was to prove the consistency of various parts of mathematics. Gödel discovered a major obstacle to achieving this goal; no reasonable list of axioms for arithmetic (i.e., the study of the natural numbers) generates a proof that those very axioms are consistent. To what extent does Gödel’s result extend beyond arithmetic to other parts of mathematics? I will describe recent work studying the reach of Gödel’s theorem in analysis (i.e., the joint study of the natural numbers and real numbers). I will not presuppose any background in mathematical logic