Part 2: BOUNDARY-NAIVE SET THEORY
Section 7: Russell Boundary Set and Validity Predicate
Pendry, S
Halfhuman Draft
January 2026
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7.1 Defining the Russell Boundary Set

Definition 7.1 (Russell Boundary Set):

R = -{ x | x ∈ x }

R is the set of all objects that do not contain themselves.

Explicit construction:

  • Let S = { x | x ∈ x } (self-containing sets)
  • Then R = U \ S
  • R contains everything except self-containing sets

The membership question:

Is R ∈ R?

Analysis:

  • R ∈ R ⟺ R ∉ S ⟺ R ∉ R (by definition of S)
  • Therefore R ∈ R ⟺ R ∉ R

Status: R exists (as -S) but has unstable self-membership. This instability is formalized through the validity predicate.

7.2 The Validity Predicate

Definition 7.2 (Validity):

For any object, statement, or construction m:

Valid(m) ⟺ m ∈ R

An object is valid if and only if it does not contain itself.

Interpretation:

  • Valid objects: Do not self-reference circularly
  • Invalid objects: Contain themselves or depend on self-reference
  • Validity is a property, not a truth value

Key insight: Validity becomes boundary-sensitive rather than absolute.

7.3 Properties of the Validity Predicate

Theorem 7.1 (Validity Non-Totality):

Not all objects can be assigned definite validity status.

Proof:

Consider R itself:

  • Valid(R) ⟺ R ∈ R
  • But R ∈ R ⟺ R ∉ R (Russell’s Paradox)
  • Therefore Valid(R) is undecidable within the system ∎

Theorem 7.2 (Self-Validation Failure):

No object can validate itself.

Proof:

Suppose object m validates itself via m ∈ m.

Then m ∉ R (since R contains only non-self-containing).

Therefore Valid(m) = FALSE.

Self-containing objects are always invalid.

Theorem 7.3 (Validity Requires External Grounding):

For Valid(m) to be determinable, assessment must reference objects other than m.

Proof:

If validity assessment for m depends only on m, then we’re checking m ∈ m.

By Theorem 7.2, this always yields invalid.

Therefore valid assessments require external reference. ∎

7.4 Consequences for Self-Reference

The validity predicate creates natural filtering:

Non-self-referencing objects:

  • Clear membership status in R
  • Valid(·) determinable
  • Behave normally in set operations

Self-referencing objects:

  • Unstable membership in R
  • Valid(·) undecidable
  • Flag themselves as requiring careful handling

Critical insight: Paradox is localized to self-referencing objects. It doesn’t propagate.

7.5 Contrast with ZFC Approach

ZFC solution:

  • Prohibit self-containing sets entirely
  • Axiom of regularity: ∀A(A ≠ ∅ → ∃x ∈ A: x ∩ A = ∅)
  • Result: R cannot be constructed

BNST solution:

  • Allow self-containing sets to exist
  • Mark them as having unstable validity
  • Result: R exists but is flagged as boundary-unstable

Advantage of BNST:

  • More permissive (allows more constructions)
  • More honest (acknowledges paradox rather than hiding it)
  • More useful (provides tools for working with self-reference)

7.6 The Separation: Existence vs. Validity

Critical distinction:

Existence: Does the set exist as a mathematical object?

Validity: Is the set’s behavior deterministic and stable?

BNST position:

  • All definable collections exist (unrestricted comprehension)
  • Not all exist validly (some have unstable boundaries)
  • Existence ≠ Validity

Analogy:

  • All numbers exist (including i)
  • Not all numbers are real (i is imaginary)
  • Existence ≠ Reality

Philosophical shift:

  • From prohibition to classification
  • From “this can’t exist” to “this exists in an unstable way”
  • From avoidance to formalization

7.7 Applications Beyond Russell’s Paradox

The validity predicate applies to any self-referencing construct:

Liar’s Paradox:

  • Statement S: “This statement is false”
  • Valid(S) ⟺ S ∈ R
  • S depends on S for truth value
  • Therefore Valid(S) = undecidable

Gödel Sentences:

  • Statement G: “G is not provable”
  • Valid(G) requires external proof system
  • Self-reference is explicit and manageable

Self-Modifying Code:

  • Program P that modifies itself
  • Valid(P) depends on whether P’s behavior is deterministic
  • Boundary-unstable if modification is self-referential

The validity predicate provides unified framework for handling self-reference across domains.

Next up
Part 2: BOUNDARY-NAIVE SET THEORY
Section 8: Conditional Complement: Validity-Gated Operations

© 2026 HalfHuman Draft - Pendry, S
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