The Omega Number: Irreducible Complexity in Pure Math

Abstract

We discuss the halting probability Ω, whose bits are irreducible mathematical facts, that is, facts which cannot be derived from any principles simpler than they are. In other words, you need a mathematical theory with N bits of axioms in order to be able to determine N bits of Ω. This pathological property of Ω is difficult to reconcile with traditional philosophies of mathematics and with traditional views of the nature of mathematical proof and of mathematical knowledge. Instead Ω suggests a quasi-empirical view of math that emphasizes the similarities between mathematics and physics rather than the differences.