Complex Systems

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

Adam Yedidia

Scott Aaronson
Massachusetts Institute of Technology
adamy@mit.edu
aaronson@csail.mit.edu

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 Zermelo–Fraenkel set theory with the axiom of choice (ZFC). 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 7910 by presenting an explicit description of a 7910-state Turing machine Z with one tape and a two-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 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.