In 1937, Quine proposed “New Foundations” set theory, and Randall Holmes has since claimed to prove its consistency using Lean. The interactive theorem prover has verified the proof, confirming the consistency of New Foundations. The project depends on mathlib, a community math library in Lean, to establish results about cardinals and groups. Tangled Type Theory (TTT) is a complex set theory with an unusual extensionality axiom, making model construction challenging. The strategy involves building a model with specific types and allowable permutations, ensuring cardinality control and satisfaction of the theory’s axioms. The project provides a model for TTT, proving the consistency of New Foundations.
https://leanprover-community.github.io/con-nf//