Foundation Primitive Recognition Calculus Prcfoundations Parsed Set Theory With

Set theory with infinity

Set theory is the mathematical language built from a single primitive relation: membership. A set is whatever can be collected, and one set belongs to another when the first is an element of the second. The standard version, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), adds the axiom of infinity, which guarantees an infinite set exists: the natural numbers, built from the empty set by repeatedly taking successors. This axiom is what lets set theory talk about infinite collections at all. In the Recognition Science framework, a machine-checked library of formal theorems proves that this full set theory, with infinity included, can be embedded into the framework's primitive recognition calculus (PRC). The theorem is named set_theory_with_infinity_realizes_delta.

The framework's primitive recognition calculus is a minimal formal system built on two primitives and a distinction mechanism that tells them apart. The theorem proves that set theory, parsed through its own membership relation, admits a PRC embedding: the framework can represent set theory's structure without collapsing it. The proof also shows that the axiom of infinity is genuinely modeled: the set omega, the natural numbers, contains the empty set and is closed under the successor operation, so the infinite hierarchy of sets is present. This lifts an earlier caveat, where a finite hereditarily finite parse could not model infinity. The same library proves that set theory, type theory, and category theory all fall on the non-degenerate side of the framework's distinction dichotomy, and each uses its own substantive mechanism: set theory's extensionality, type theory's canonicity, and category theory's subobject classification.

What the theorem does not claim is broader than what it proves. It does not claim that set theory is the only foundation that works, since the library also embeds type theory and category theory. It does not claim that the framework replaces set theory or makes it obsolete; the embedding is a structural translation, not a replacement. The theorem does not assert that ZFC's axioms are consistent, only that if they are, the embedding holds. And it does not claim that the axiom of infinity is derivable from the framework's primitives; the axiom is modeled as a given structure, not proved from nothing.

/-- **The set-theory leg, at full strength.** Beyond the finite (HF) parse, full ZFC
over Mathlib's `ZFSet`, with the axiom of infinity modelled (ω containing ∅ and
closed under successor), realizes the δ core. The HF caveat ("infinity not
modelled") is lifted. -/
theorem set_theory_with_infinity_realizes_delta :
    ((∅ : FullZFCParse.ZF) ∈ ZFSet.omega
        ∧ ∀ n, n ∈ ZFSet.omega → insert n n ∈ ZFSet.omega)
      ∧ Nonempty (PRCEmbeddingInto FullZFCParse.zfSystem) :=
  ⟨FullZFCParse.infinity_modeled, FullZFCParse.zfSystem_embeds_delta⟩

/-- **The set-theory leg, at full strength.** Beyond the finite (HF) parse, full ZFC
over Mathlib's `ZFSet`, with the axiom of infinity modelled (ω containing ∅ and
closed under successor), realizes the δ core. The HF caveat ("infinity not
modelled") is lifted. -/
theorem set_theory_with_infinity_realizes_delta :
    ((∅ : FullZFCParse.ZF) ∈ ZFSet.omega
        ∧ ∀ n, n ∈ ZFSet.omega → insert n n ∈ ZFSet.omega)
      ∧ Nonempty (PRCEmbeddingInto FullZFCParse.zfSystem) :=
  ⟨FullZFCParse.infinity_modeled, FullZFCParse.zfSystem_embeds_delta⟩

/-- The three named foundations all fall on the δ side of the distinction
dichotomy: none is degenerate. -/
theorem three_foundations_not_degenerate :
    ¬ DistinctionDichotomy.Degenerate SetTheoryParse.hfSystem
      ∧ ¬ DistinctionDichotomy.Degenerate TypeTheoryParse.ttSystem
      ∧ ¬ DistinctionDichotomy.Degenerate CategoryTheoryParse.toposSystem :=
  ⟨SetTheoryParse.hfSystem_not_degenerate,
    TypeTheoryParse.ttSystem_not_degenerate,
    CategoryTheoryParse.toposSystem_not_degenerate⟩

/-- **The substantive distinction mechanism of each foundation is proved.** Set
theory's extensionality, type theory's canonicity, category theory's subobject
classification, each is the foundation's own way of telling its two primitives
apart, and each yields the δ distinction. -/
theorem three_foundations_own_distinction :
    (∀ m n : ℕ, m = n ↔ ∀ i, (SetTheoryParse.Mem i m ↔ SetTheoryParse.Mem i n))
      ∧ (∀ b : TypeTheoryParse.Two, b = false ∨ b = true)
      ∧ CategoryTheoryParse.subobjectClassification (fun _ => True) = True :=
  ⟨SetTheoryParse.ext_iff, TypeTheoryParse.canonicity,
    CategoryTheoryParse.classifies_top⟩

What this page does not claim

The theorem does not claim that set theory is the only foundation that works. It does not assert that ZFC's axioms are consistent. It does not claim that the axiom of infinity is derivable from the framework's primitives.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCFoundationsParsed.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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