Foundation Primitive Recognition Calculus Prcset Theory Parse Not Mem Empty

The empty set

The empty set is the set with no members. In the Recognition Science framework's coding of hereditarily finite sets, each set is represented by a natural number, and membership is read off the binary digits of that number: a code i is a member of code n exactly when the i-th bit of n is set. This is the standard Ackermann coding. The empty set is coded by the number 0, whose binary representation has no bits set at all.

The theorem not_mem_empty states this fact formally: for every natural number i, it is not the case that i is a member of 0. The proof is a single simplification step, since the definition of membership immediately reduces to checking a bit of 0, which is always false. This is not a deep result, but it is a necessary foundation stone: it pins down the bottom of the set-theoretic hierarchy, the object from which all other sets are built.

In Recognition Science, this theorem is part of a larger construction. The framework models a formal system whose tokens are set codes, whose discrimination relation is extensional set inequality, and whose endpoints are the empty set and its singleton. The theorem not_mem_empty is one of the five properties that together show this system realizes the framework's δ core, a minimal structure of distinction. It also appears in the packaged theorem hf_set_theory_realizes_delta, which bundles extensionality, the empty set property, the singleton property, and the discrimination relation into a single statement.

The theorem does not claim that the empty set is unique, that it is the only set with no members, or that the framework's coding is the only way to represent sets as numbers. It only establishes the local fact about the number 0 in this particular coding. The broader significance, that this coding satisfies extensionality and realizes the δ core, is established by separate theorems in the same file.

/-- The empty set is coded by `0`: it has no members. -/
theorem not_mem_empty (i : ℕ) : ¬ Mem i 0 := by
  simp [Mem]

/-- The empty set is coded by `0`: it has no members. -/
theorem not_mem_empty (i : ℕ) : ¬ Mem i 0 := by
  simp [Mem]

/-- **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 theorem does not claim that the empty set is unique among all sets with no members. The theorem does not claim that the Ackermann coding is the only way to represent sets as natural numbers. The theorem does not claim that the framework's set theory is a complete foundation for all of mathematics.

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)

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