Aaronson’s unprovability conjecture for Busy Beaver values

Determine whether the Busy Beaver values S(20) and S(10) are respectively unprovable in Zermelo–Fraenkel set theory (ZF) and Peano Arithmetic (PA), i.e., establish that ZF does not prove the exact value of S(20) and PA does not prove the exact value of S(10).

Background

The paper surveys the deep connections between Busy Beaver values and the halting problem, noting that many Π0_1 statements (including famous conjectures and consistency claims) can be encoded as halting problems for small Turing machines. This leads to questions about the provability of specific Busy Beaver values within common axiomatic systems.

In this context, the authors cite Aaronson’s conjecture that surprisingly small Busy Beaver values might already be independent of standard foundations (ZF and PA), underscoring the potential logical limits of proving such values.

References

Aaronson conjectures that as low as $S(20)$ cannot be proved in ZF and $S(10)$ cannot be proved in PA .

— Determination of the fifth Busy Beaver value  (2509.12337 - Collaboration et al., 15 Sep 2025) in Introduction — Challenges