A relatively small turing machine whose behavior is independent of set theory

Abstract

Since the definition of the Busy Beaver function by Radó in 1962, an interesting open question has been what the smallest value of n for which BB(n) is independent of ZFC. Is this n approximately 10, or closer to 1,000,000, or is it unfathomably large? In this thesis, I show that it is at most 340,943 by presenting an explicit description of a 340,943-state Turing machine Z with 1 tape and a 2-symbol alphabet whose behavior cannot be proved in ZFC, assuming ZFC is consistent. The machine is based on work of Harvey Friedman on independent statements involving order-invariant graphs. Ill In doing so, I give the first known upper bound on the highest provable Busy Beaver number in ZFC. I also present an explicit description of a 7,902-state Turing machine G that halts if and only if there's a counterexample to Goldbach's conjecture, and an explicit description of a 36,146-state Turing machine R that halts if and only if the Riemann hypothesis is false. In the process of creating G, R, and Z, I define a higher-level language, TMD, which is much more convenient than direct state manipulation, and explain in great detail the process of compiling this language down to a Turing machine description. TMD is a well-documented language that is optimized for parsimony over efficiency. This makes TMD a uniquely useful tool for creating small Turing machines that encode mathematical statements.