Lean's type theory proves Con(ZF) from excluded middle alone — result found with AI
MikePFrank · x · 2026-09-23
Mario Carneiro's arXiv paper shows that Lean's core type theory, with classical excluded middle and no choice axiom, is strong enough to prove the consistency of ZF set theory outright, with the proof fully formalized in Lean. This overturns the expectation that without a choice/description operator the theory falls well below ZF. perrymetzger notes the result was found with AI; Mike Frank pushes back that it only implies Con(ZF) → Con(Lean) and that Lean has had known bugs allowing False to be proven. The mechanism uses large elimination of an accessibility predicate over a well-founded tree guarded by propositions.
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