Modal Theory of the Category of Sets
Semantic Grz.2 Classification — Formalized in Lean 4
Formal Verification Certificate
Every theorem in this project has been machine-checked by the Lean 4 kernel. No axiom is assumed without proof. No gap exists in the verification chain.
Modal Theory of the Category of Sets • Lean 4 + Mathlib • Apoth3osis Labs
Key Mathematics
FormulaTheory.Grz2 — the semantic characterization of the modal logic Grz.2 as a formula theory
grz2_iff_validAtBottomAllFiniteJoinSemilattices
infsets_formulaic_grz2_upper_bound_at_nat_root — classification via all-functions substitution at natural number root
infsurj_formulaic_grz2_upper_bound_at_nat_root — classification via surjections substitution at natural number root
SEMANTIC BOUNDARY
This formalization internalizes semantic as the class of formulas valid on all finite Boolean frames. The classification is frame-semantic: the space of valid formulas is characterized by finite-frame conditions and infinite-world partition elimination. It does not claim a full internal Hilbert calculus or syntactic completeness proof for Grz.2. The individual axioms T, 4, .2, and Grz are proved to belong to the semantic theory, but derivability in a proof system is not formalized.
Proof Blueprint
Verified Lean Artifacts
This project ships as a Lean-first artifact package. All 37 source modules are available for download and independent verification.
Resources
“Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.”
Frank Herbert, Dune
A janitor who proves a theorem outranks a tenured professor who publishes noise.
Not as a slogan. As a structural fact of how the network operates. The only currency that matters is the quality of your contribution, measured not by committee but by mathematics.
A valid, original framing or conjecture
Formal argument with paper-level rigor
Connecting theory to observable outcomes
Working software the project depends on
Reproducible research with methodology and data
Machine-verified claim checked by a proof assistant
Foundational, load-bearing implementation
Connecting subsystems or knowledge domains end-to-end
Strategic direction & roadmaps
Writing, documentation & papers
Visual, UX & information design
Teaching, tutorials & workshops
Community, outreach & events
Partnerships, governance & policy
Translation, media & accessibility
Ethics, review & quality assurance
Every accepted contribution receives a MENTAT Contribution Record — cryptographically signed, IPFS-pinned, permanently yours. No committee decides your worth. The type checker does.