Results Glossary Entry Canonical mathematics K6 is the closure axiom of the τ-kernel: there is no sixth generator. Any candidate sixth atom must reduce to a finite combination of K1–K5. K6 is what makes the τ-kernel finitely generated and what makes the categoricity theorem (T03 / I.T…
Results · Mathematics Glossary · Postulate MathG-K03-no-omega-axiom K6 Canonical

The no-ω axiom (K6)

K6 is the closure axiom of the τ-kernel: there is no sixth generator. Any candidate sixth atom must reduce to a finite combination of K1–K5. K6 is what makes the τ-kernel finitely generated and what makes the categoricity theorem (T03 / I.T08) provable.

Mathematics Glossary Primary: I.K6

τ-Definition

K6 is the closure axiom of the τ-kernel: there is no sixth generator. Any candidate sixth atom must reduce to a finite combination of K1–K5. K6 is what makes the τ-kernel finitely generated and what makes the categoricity theorem (T03 / I.T08) provable.

Categorical invariant. A meta-logical constraint that forbids extending the kernel signature beyond K1–K5; expressed as the assertion that any τ-categorical invariant is reachable by finite K1–K5 composition.

Primary registry anchor: I.K6

Supporting items: I.K0, I.K1, I.T08

τ-Derivation Chain

  1. I.K0 — Universe Postulate — kernel τ exists
  2. I.K1 — Five canonical generators K1–K5 are posited
  3. I.K6 — No-ω axiom — no sixth generator can be added without contradicting closure

Lean modules referenced: TauLib.BookI.MetaLogic.OnticInvariance

Mathematical content

Axiom K6
Axiom

Every τ-categorical morphism is reachable by finite composition from K1–K5; equivalently, no candidate sixth generator is independent of K1–K5.

Role. closure-meta-logical

Why "no ω". Without K6, the kernel could be extended by an ω-th independent generator, producing non-isomorphic models of K0 + K1–K5. K6 forbids such extensions; together with K0 + K1–K5 it implies categoricity (T03).

Consequence. K6 is what makes the τ-framework's count-of-axioms compact: the kernel is exactly six axioms (K1–K6) plus the existence postulate (K0).

Lean Coverage

Status: Formalized

Module: TauLib.BookI.MetaLogic.OnticInvariance

Lean kind: axiom

Lean symbol: Tau.BookI.MetaLogic.noOmega

Cross-domain bridges

This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.

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