Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Expressi

The expressive system

The hereditarily finite sets are the sets built from the empty set by finitely many steps of forming finite collections. They are the universe of finite mathematics: everything from the natural numbers to the finite ordinals lives inside them. The framework's machine-checked library of formal theorems encodes each such set as a natural number, using a standard binary coding where the members of a set are the positions of the 1-bits in its code. This is the Ackermann coding, and it makes the membership relation a concrete operation on ordinary numbers.

In this coding, the empty set gets code 0, and the set containing the empty set gets code 1. The library proves that two codes are equal exactly when they have the same members, which is the axiom of extensionality. It proves that code 0 has no members, and that code 1 has exactly one member, namely code 0. The discrimination relation of the system is genuine set inequality: two codes are distinguished exactly when they differ in some member, not by an accident of their binary representation.

The declaration hfSystem_expressive establishes one specific fact: the system built from these codes is expressive. In the framework's vocabulary, an expressive system is one whose two endpoints, here the empty set and its singleton, are distinct. The proof is a direct computation showing that 0 and 1 are not equal as natural numbers. This single fact starts a chain of consequences: the library derives that this system embeds the framework's δ core, that it is not degenerate, and that it falls on the δ side of the distinction dichotomy. The packaged theorem hf_set_theory_realizes_delta bundles all of these properties into one statement.

What the declaration does not claim is larger than what it does. It does not claim that the framework's recognition ledger, a discrete record of events, is the same thing as the hereditarily finite sets. It shows that set theory can be parsed into the framework's formal system interface, not that sets are the ledger. It does not claim that all of set theory, including infinite sets or the axioms of choice and replacement, is captured here. The coding covers only the hereditarily finite sets, the finite part of the universe. It does not claim that the framework derives the axioms of set theory from nothing; the extensionality and membership facts are proved from the coding definition, not assumed as primitive.

The practical consequence is that the framework's foundational claims have a concrete, checkable substrate. When the framework speaks of tokens, expressions, and traces, the hereditarily finite sets provide a realized example where the definitions are not vacuous. The reader can see that the framework's formal system interface is not an empty abstraction: it has at least one substantial model, and that model is the ordinary finite mathematics that every working mathematician already uses.

/-- `hfSystem` distinguishes its endpoints: ∅ ≠ {∅}. -/
theorem hfSystem_expressive : hfSystem.Expressive := by
  show (0 : ℕ) ≠ 1
  decide

/-- **Extensionality.** Two HF codes are equal iff they have the same members. The
Ackermann interpretation satisfies the axiom of extensionality; it is exactly ℕ bit
extensionality. -/
theorem ext_iff (m n : ℕ) : m = n ↔ ∀ i, (Mem i m ↔ Mem i n) := by
  refine ⟨fun h i => by rw [h], fun h => Nat.eq_of_testBit_eq fun i => ?_⟩
  have hi := h i
  cases hm : Nat.testBit m i <;> cases hn : Nat.testBit n i <;> simp_all [Mem]

/-- The discrimination relation IS the foundation's own extensional set inequality:
two codes are distinguished exactly when they differ in some member. So `hfSystem`
discriminates by genuine set difference, not by code accident. -/
theorem distinguishes_iff_extensional (a b : ℕ) :
    hfSystem.distinguishes a b ↔ ∃ i, ¬ (Mem i a ↔ Mem i b) := by
  show a ≠ b ↔ ∃ i, ¬ (Mem i a ↔ Mem i b)
  rw [ne_eq, ext_iff a b]
  push_neg
  rfl

/-- **HF set theory contains the δ core.** -/
theorem hfSystem_embeds_delta : Nonempty (PRCEmbeddingInto hfSystem) :=
  FormalSystemEmbeddingTarget_proved hfSystem hfSystem_expressive

/-- HF set theory falls on the δ side of the distinction dichotomy: it is
non-degenerate, hence realizes δ. -/
theorem hfSystem_not_degenerate : ¬ DistinctionDichotomy.Degenerate hfSystem :=
  DistinctionDichotomy.not_degenerate_of_realizesDelta hfSystem hfSystem_embeds_delta

/-- **The faithful parse, packaged.** HF set theory, encoded by Ackermann coding,
(i) satisfies extensionality, (ii) has ∅ = code 0 with no members, (iii) has
{∅} = code 1 with exactly the member ∅, (iv) discriminates by genuine set
difference, and (v) realizes the δ core. The endpoints ∅ and {∅} are the von
Neumann 0 and 1, distinguished as sets. -/
theorem hf_set_theory_realizes_delta :
    (∀ m n : ℕ, m = n ↔ ∀ i, (Mem i m ↔ Mem i n))
      ∧ (∀ i, ¬ Mem i 0)
      ∧ (∀ i, Mem i 1 ↔ i = 0)
      ∧ (∀ a b : ℕ, hfSystem.distinguishes a b ↔ ∃ i, ¬ (Mem i a ↔ Mem i b))
      ∧ Nonempty (PRCEmbeddingInto hfSystem) :=
  ⟨ext_iff, not_mem_empty, mem_one_iff, distinguishes_iff_extensional,
    hfSystem_embeds_delta⟩

What this page does not claim

The hereditarily finite sets are the same thing as the framework's recognition ledger. The coding captures infinite sets or the full axioms of set theory. The framework derives the axioms of set theory from nothing; the facts are proved from the coding definition.

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/PRCSetTheoryParse.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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