Foundation Primitive Recognition Calculus Prcset Theory Parse Hf System Expr Ref

A reflexive order

The hereditarily finite sets are the sets built from the empty set by finitely many steps of collecting previously built sets: ∅, {∅}, {∅, {∅}}, and so on. A classical fact about them is that they can be coded as natural numbers, for instance by the Ackermann coding in which a number's binary bits list its members. The declaration hfSystem_exprReflexive proves a small structural property of this coding: the order relation used to compare expressions is reflexive, meaning every expression extends itself.

In the framework's language, an expression is a code for a set, and one expression extends another when its code is at least as large in the natural ordering of numbers. The theorem states that this relation holds for every code: each number is at least itself. The proof is one line, because the property is exactly the reflexivity of ≤ on the natural numbers, a fact the machine-checked library of formal theorems records as Nat.le_refl.

The declaration does not claim that the order is antisymmetric, transitive, or total. It does not say that every set is a member of itself, which would contradict the axiom of foundation. It only asserts the weakest form of comparability: a set stands in the extension relation to itself. This reflexivity is a prerequisite for the framework's distinction dichotomy, which sorts formal systems into degenerate and non-degenerate cases, but the declaration itself does not establish which side the hereditarily finite sets fall on.

What the reflexivity does establish is that the coding is coherent as a basis for further analysis. It guarantees that the formal system built on these codes has a well-defined notion of expression extension, without which the framework's recognition calculus could not proceed. The property is a foundation stone, not a finished building.

theorem hfSystem_exprReflexive : DistinctionDichotomy.ExprReflexive hfSystem :=
  fun n => Nat.le_refl n

What this page does not claim

The declaration does not claim that the order is antisymmetric, transitive, or total. It does not claim that any set is a member of itself. It does not establish which side of the distinction dichotomy the hereditarily finite sets fall on.

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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