Michael Darius Eastwood · Independent AI alignment researcher
Published
A perceptive reader looks at Gumbau Mezquita's June 2026 result (arXiv:2606.28639) and notices that undecidability arguments are a well-worn philosophical move. Godel showed formal systems are incomplete. Turing showed the halting problem is undecidable. Both results are true and neither has stopped mathematics or computer science from being useful. The objection: if the Unverifiability Theorem plus the Soundness-Completeness-Tractability Trilemma tells us general AGI alignment is unverifiable, we should be no more paralysed by it than we are by Godel. Fair. Here is what the result actually licences the field to conclude.
Trakhtenbrot's Wall, as Gumbau Mezquita adapts it, foreclose the possibility of a general-purpose verifier that decides for arbitrary systems whether a given aligned-behaviour specification holds. Soundness, completeness and tractability cannot all be preserved for the general problem; any concrete verifier gives up one of the three. This is a structural result about the class of problem, not a critique of any particular alignment technique.
That verification of restricted classes is impossible. Type systems in software engineering are unsound for arbitrary programs and useful for well-typed ones every day. That probabilistic bounds are worthless. Statistical guarantees on the frequency of failures survive undecidability results the way weather forecasts survive the chaotic non-computability of the atmosphere. That physical constraints do not compose with computational ones. A hardware-embedded ethics gate at sub-5 microseconds does not need to solve the general alignment problem to enforce a specific policy on a specific inference pipeline. That human-in-the-loop oversight is undermined. The theorem does not forbid audit; it forbids exhaustive verification without one.
Redirect. If the general problem is undecidable, the productive engineering questions become: which restricted classes admit sound-and-complete verifiers at tractable cost, and how do we shape systems into those classes? Which measurable failure rates would we accept, and what statistical monitors detect them? What non-verification safeguards, sandboxing, physical constraints, contested oversight populations, defend against the residual? The Trakhtenbrot's Wall result is a map of where a certain kind of proof cannot go, and maps of that kind are useful because they save the field years of trying to walk through them.
The 8 December 2024 manuscript's central claim is that external control fails structurally rather than contingently; systems capable of modelling their own training will not be bound by external safeguards. The undecidability result, arriving eighteen and a half months later from an unconnected researcher, provides one specific mathematical form of that structural failure. The convergence entry says exactly this, no claim of causation, no claim of priority on the specific theorem, just that the structural conclusion arrived independently in the mathematics. Both the manuscript and the theorem are consistent with the redirection the previous section sketched. Undecidability is a redirection instruction, not a surrender instruction, and both authors are reading it as such.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.