What is the Universal Forcing theorem?

permalink: https://pith.science/recognition/ask/what-is-the-universal-forcing-theorem-e8ec975b · first asked 2026-07-03 · browse all asks

confidence: high in recognition cached

Setting: admissible Law-of-Logic realizations

The setting consists of any two Law-of-Logic realizations R and S, each supplying a carrier, cost, identity-step data, and orbit satisfying the structural laws.

Theorem: same arithmetic structure forced across realizations

The theorem asserts that the forced arithmetic objects are canonically equivalent: (arithmeticOf R).peano.carrier ≃ (arithmeticOf S).peano.carrier, as both are initial Peano algebras.

Cited Lean anchors

universal_forcing states the equivalence; it is defined using arithmeticOf and arithmetic_invariant.

cited recognition theorems

UniversalForcing.universal_forcing MUST anchor: direct statement of the Universal Forcing theorem as equivalence of forced arithmetic carriers.
UniversalForcing.arithmeticOf SHOULD anchor: defines the forced arithmetic object extracted from a realization.
UniversalForcing.arithmetic_invariant SHOULD anchor: states the canonical equivalence between arithmetic objects of two realizations.

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.