IndisputableMonolith.Foundation.UniversalForcingSelfReference
This module introduces the meta-carrier as the type of LogicRealization.{0,0} instances sitting in Type 1, together with an associated metaCost function and MetaRealizationCert certificates. It equips the universal forcing construction with self-referential structure on top of the imported LogicRealization and NaturalNumberObject interfaces. Researchers formalizing categorical logic or initial algebras would cite it when lifting forcing results to the meta level. The module is built from definitions and elementary lemmas on cost symmetry and ar
claimLet $M$ be the type of all LogicRealization instances (in Type 1). Define metaCost : $M$ → ℕ satisfying metaCost_self, metaCost_symm and metaCost_total, together with MetaRealizationCert as the inhabited certificate type witnessing metaForcedArithmeticInvariance.
background
The module imports LogicRealization, which supplies the setting-independent interface into which continuous, discrete and categorical realizations of the law of logic are mapped. UniversalForcing asserts that any two such realizations yield canonically equivalent forced arithmetic objects because those objects are initial Peano algebras. NaturalNumberObject supplies the Lawvere triple (N, z, s) characterization of the forced naturals. The present module adds the meta-carrier layer so that these objects can refer to their own realizations.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the meta-level carrier and cost structure required by the self-reference step of the Universal Forcing theorem. It feeds the canonical equivalence of forced arithmetic objects across realizations and closes the loop from LogicRealization to the natural-number object. No open scaffolding remains inside the module.
scope and limits
- Does not contain the full statement or proof of the Universal Forcing theorem.
- Does not construct explicit continuous or discrete realizations.
- Does not address higher universes beyond Type 1 for the meta-carrier.
- Does not derive numerical values for metaCost on concrete realizations.
depends on (4)
declarations in this module (14)
-
structure
for -
abbrev
MetaCarrier -
def
metaCost -
theorem
metaCost_self -
theorem
metaCost_symm -
theorem
metaCost_total -
theorem
metaCost_eq_zero_iff -
def
metaForcedArithmeticInvariance -
theorem
metaForcedArithmeticInvariance_self -
structure
MetaRealizationCert -
def
metaRealizationCert -
theorem
metaRealizationCert_inhabited -
theorem
framework_is_reflexively_closed -
theorem
meta_meta_theorem