A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory

TL;DR

This paper constructs a small Turing machine with 7,910 states that cannot be proven to halt within ZFC, providing an explicit upper bound on the smallest such machine whose behavior is independent of set theory.

Contribution

It presents the first explicit upper bound on the minimal size of a Turing machine with behavior independent of ZFC, using a novel higher-level language Laconic for machine design.

Findings

A 7,910-state Turing machine is independent of ZFC assuming consistency.

Developed Laconic language for easier Turing machine construction.

Created machines G and R linked to Goldbach's Conjecture and Riemann Hypothesis.

Abstract

Since the definition of the Busy Beaver function by Rado in 1962, an interesting open question has been the smallest value of n for which BB(n) is independent of ZFC set theory. Is this n approximately 10, or closer to 1,000,000, or is it even larger? In this paper, we show that it is at most 7,910 by presenting an explicit description of a 7,910-state Turing machine Z with 1 tape and a 2-symbol alphabet that cannot be proved to run forever in ZFC (even though it presumably does), assuming ZFC is consistent. The machine is based on the work of Harvey Friedman on independent statements involving order-invariant graphs. In doing so, we give the first known upper bound on the highest provable Busy Beaver number in ZFC. To create Z, we develop and use a higher-level language, Laconic, which is much more convenient than direct state manipulation. We also use Laconic to design two Turing machines, G and R, that halt if and only if there are counterexamples to Goldbach's Conjecture and the Riemann Hypothesis, respectively.