Part 2: BOUNDARY-NAIVE SET THEORY
Section 5: Naive Set Theory Baseline
Pendry, S
Halfhuman Draft
January 2026
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5.1 Classical Naive Set Theory

Before examining our extensions, we establish the baseline: classical naive set theory as it existed before Russell’s discovery.

Fundamental Principle:

A set is any definable collection. If you can describe a property P(x), then { x | P(x) } exists as a set.

Core Operations:

Membership: x ∈ A (x is an element of A)

Subset: A ⊆ B ⟺ ∀x(x ∈ A → x ∈ B)

Union: A ∪ B = { x | x ∈ A ∨ x ∈ B }

Intersection: A ∩ B = { x | x ∈ A ∧ x ∈ B }

Complement: A’ = { x | x ∉ A } (relative to some universal set)

Key Properties:

Unrestricted Comprehension: { x | P(x) } exists for any property P

Universal Set: U = { x | x = x } contains everything

Empty Set: ∅ = { x | x ≠ x } contains nothing

Self-Membership Allowed: A ∈ A is syntactically valid

5.2 Why Naive Set Theory Failed

Russell’s Paradox (1901):

Define: R = { x | x ∉ x } (sets that don’t contain themselves)

Question: Is R ∈ R?

Case 1: Suppose R ∈ R

  • Then R satisfies the condition x ∉ x
  • Therefore R ∉ R
  • Contradiction

Case 2: Suppose R ∉ R

  • Then R satisfies the condition x ∉ x
  • Therefore R ∈ R (by definition of R)
  • Contradiction

Conclusion: R ∈ R ⟺ R ∉ R (contradiction)

5.3 The Traditional Solution: Restriction

Zermelo-Fraenkel approach:

Axiom of Regularity (Foundation):

∀A(A ≠ ∅ → ∃x ∈ A: x ∩ A = ∅)

Consequence: No set contains itself (A ∉ A)

Replacement Schema: Limits comprehension not every property defines a set

Result: Russell’s set R cannot be constructed within ZFC

Cost: Loss of unrestricted comprehension, universal sets, intuitive simplicity

5.4 Alternative: Formalization Instead of Prohibition

Our approach: Keep naive set theory intact but add tools to handle paradox

Key insight: Russell’s Paradox doesn’t prove R doesn’t exist it proves R exists in a contradictory way that needs formal handling.

Analogy:

  • i² = -1 doesn’t prove i doesn’t exist
  • It defines how i behaves
  • Similarly, R ∈ R ⟺ R ∉ R might define how R behaves

5.5 What We Preserve from Naive Set Theory

BNST explicitly maintains:

Unrestricted Comprehension:

∀P: ∃S: ∀x(x ∈ S ↔ P(x))

Universal Set:

∃U: ∀x(x ∈ U)

Self-Membership Allowed:

A ∈ A is syntactically and semantically valid

No Artificial Restrictions:

No axiom of regularity

No type hierarchy

No stratification requirements

5.6 What We Add to Naive Set Theory

Three new primitives:

1. Boundary Complement Operator (-{}):

Formal tool for negation and exclusion

2. Validity Predicate (Valid(·)):

Distinguishes self-referencing from externally-grounded

3. Conditional Complement Rule:

Operators themselves must satisfy validity

Goal: Preserve naive set theory’s expressiveness while preventing logical explosion from paradox

The following sections develop each component formally.

Next up
Part 2: BOUNDARY-NAIVE SET THEORY
Section 6: The Boundary Complement Operator

© 2026 HalfHuman Draft - Pendry, S
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