What is the Universal Forcing theorem?

permalink: https://pith.science/recognition/ask/what-is-the-universal-forcing-theorem-e8ec975b · first asked 2026-08-19 · browse all asks

confidence: high in recognition cached

Setting: admissible Law-of-Logic realizations

The setting consists of admissible Law-of-Logic realizations, which are instances of LogicRealization providing a carrier with comparison cost, identity-step data, and an orbit satisfying the structural laws.

Theorem: same arithmetic structure forced across realizations

The theorem states that any two Law-of-Logic realizations force canonically equivalent arithmetic objects, specifically that the Peano carriers extracted from them are equivalent via the unique map between initial Peano algebras.

Cited Lean anchors

The result is established by universal_forcing, which relies on arithmeticOf and arithmetic_invariant.

cited recognition theorems

UniversalForcing.universal_forcing Main theorem establishing canonical equivalence of forced arithmetic across any two LogicRealization instances.
UniversalForcing.arithmeticOf Extracts the arithmetic object from a given LogicRealization.
UniversalForcing.arithmetic_invariant Gives the invariant form of the equivalence between arithmetic objects.

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.