Universal Set Ontology

FLOW ALGEBRA
UNIVERSAL SET ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Set Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/set-entities/universal-set-ontology-3.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13
Status: Formal ontology specification based on established set theory; source page not independently verified.

01. PURPOSE
flow "UniversalSetOntology" v1.0
  domain: set_theory
  entity: UniversalSet
  intent: represent the ambient set of all objects under a specified discourse
  context_required: true

02. CORE DEFINITIONS
  Universe(U, C) := U is the collection of admissible objects in context C
  SetInContext(A, C) => A subseteq U(C)
  x in U(C) <=> Admissible(x, C)
  EmptySet = {}
  U(C) != EmptySet is NOT required; empty universes may be allowed by the model
  NOTE: A universal set is relative to a discourse or typed domain.
  NOTE: Unrestricted 'set of all sets' is inconsistent in ordinary ZFC set theory.

03. TYPES
  Context: identifier
  Element: typed entity
  Set: collection of elements
  UniversalSet: Set with a context and membership criterion
  Predicate: Element -> Boolean
  Relation: typed relation among entities

04. ENTITY SCHEMA
  UniversalSet {
    id: URI
    label: string
    context_id: Context
    element_type: Type
    membership_predicate: Predicate
    explicit_members: Set<Element>?
    cardinality: Cardinal?
    closed_world: Boolean
    provenance: URI?
    confidence: [0,1]
    version: string
  }

05. RELATIONSHIPS
  UniversalSet --definesDomainFor--> Context
  Element --memberOf--> UniversalSet
  Set --subsetOf--> UniversalSet
  UniversalSet --hasElementType--> Type
  UniversalSet --providesComplementDomainFor--> Set
  UniversalSet --constrainsQuantifier--> LogicalStatement
  UniversalSet --hasCardinality--> Cardinal

06. SET INCLUSION
  forall A: SetInContext(A,C) => A subseteq U(C)
  EmptySet subseteq U(C)
  U(C) subseteq U(C)
  A subseteq U(C) <=> forall x (x in A => x in U(C))

07. UNION AND INTERSECTION
  A union U(C) = U(C), provided A subseteq U(C)
  A intersection U(C) = A, provided A subseteq U(C)
  EmptySet union U(C) = U(C)
  EmptySet intersection U(C) = EmptySet

08. COMPLEMENT
  Complement_C(A) := U(C) \\ A
  x in Complement_C(A) <=> x in U(C) and x notin A
  Complement_C(U(C)) = EmptySet
  Complement_C(EmptySet) = U(C)
  Complement_C(Complement_C(A)) = A, if A subseteq U(C)
  Complement_C(A union B) = Complement_C(A) intersection Complement_C(B)
  Complement_C(A intersection B) = Complement_C(A) union Complement_C(B)

09. DIFFERENCE AND SYMMETRIC DIFFERENCE
  U(C) \\ A = Complement_C(A)
  A \\ U(C) = EmptySet, if A subseteq U(C)
  A symmetricDifference U(C) = Complement_C(A), if A subseteq U(C)

10. POWER SET
  PowerSet(U(C)) := {A | A subseteq U(C)}
  A in PowerSet(U(C)) <=> A subseteq U(C)
  EmptySet in PowerSet(U(C))
  U(C) in PowerSet(U(C))
  For finite |U(C)| = n: |PowerSet(U(C))| = 2^n
  For every set U(C): |PowerSet(U(C))| > |U(C)| (Cantor theorem)

11. CARDINALITY
  finite U: cardinality(U) = number of distinct members
  infinite U: cardinality(U) is an infinite cardinal
  A subseteq U => |A| <= |U|
  A subseteq U does not imply |A| < |U| for infinite U

12. QUANTIFICATION
  forall x in U(C): P(x) := forall x (x in U(C) => P(x))
  exists x in U(C): P(x) := exists x (x in U(C) and P(x))
  Quantifier meaning depends on context C.

13. CONTEXT AND SCOPE
  U(NaturalNumbers) = N
  U(RealNumbers) = R
  U(EmployeesOfOrganisation) = set of employees satisfying the declared scope
  Different contexts may yield different universal sets.
  Do not silently equate U(C1) and U(C2).

14. CLASSICAL LOGIC VS OPEN WORLD
  Under closed-world semantics, absence from an exhaustive U can imply nonmembership.
  Under open-world semantics, missing membership evidence does not imply nonmembership.
  closed_world flag must be explicit.

15. SEMANTIC EQUIVALENCE
  EquivalentUniverse(U1,U2) iff same typed membership extension under compatible contexts
  Label similarity alone is insufficient.
  Context equivalence must be validated before universe equivalence.
  U1 = U2 => forall x (x in U1 <=> x in U2)
  The converse requires both objects to be sets in an extensional set theory.

16. INFERENCE RULES
  R1: A subseteq U and x in A => x in U
  R2: x in U and x notin A => x in Complement_C(A), if A subseteq U
  R3: A subseteq U => A intersection U = A
  R4: A subseteq U => A union U = U
  R5: A subseteq U => Complement_C(Complement_C(A)) = A
  R6: A subseteq U and B subseteq A => B subseteq U

17. VALIDATION
  require context_id and element_type
  require all declared members satisfy membership_predicate
  require all context-scoped sets are subsets of U(C)
  reject complement without specified universe
  reject universal-set claims that contradict the declared set theory
  distinguish unknown membership from false membership
  reject inconsistent cardinality claims

18. EDGE CASES
  U(C) = EmptySet is valid where empty domains are permitted.
  In first-order logic with standard nonempty-domain semantics, U(C) must be nonempty.
  Relative complements depend on U(C).
  Infinite universes need not be enumerable.
  Russell paradox prevents an unrestricted universal set in standard ZFC.
  Alternative foundational systems may handle universal collections differently.

19. KNOWLEDGE GRAPH MAPPING
  node: U(C) type UniversalSet
  edge: (U(C), definesDomainFor, C)
  edge: (A, subsetOf, U(C))
  edge: (x, memberOf, U(C)) only when membership is established
  edge: (U(C), hasElementType, T)
  provenance attaches to each claim and context declaration

20. CONFIDENCE
  confidence = 1.0 for formally verified axioms and derivations in declared logic
  confidence < 1.0 for extracted or inferred contextual membership claims
  confidence must not replace proof status
  distinguish asserted, derived, verified and disputed claims

21. FLOW_HASH
  canonicalize: ontology_id, version, context, types, axioms, rules
  exclude: volatile timestamps and presentation-only whitespace
  FLOW_HASH := SHA256(canonical_serialization)
  Recompute after semantic changes.

22. UPDATE_MODEL
  on change of membership criterion: revalidate members and subset edges
  on context change: re-evaluate quantifiers and complements
  on new element: check admissibility and downstream inferences
  on ontology revision: increment version and recompute FLOW_HASH
  preserve provenance and change history

23. AI_CONSUMPTION
  parse context before evaluating membership
  preserve type information
  use context-relative complements only
  distinguish set universe from logical domain and proper class
  apply inference rules only when premises and logic assumptions hold
  return derivations with evidence and confidence
  never infer unrestricted universal set existence in ZFC

24. TEST CASES
  C = integers from 1 through 5; U = {1,2,3,4,5}
  A = {2,4}
  assert A subseteq U
  assert Complement_C(A) = {1,3,5}
  assert A union U = U
  assert A intersection U = A
  assert Complement_C(U) = EmptySet
  assert cardinality(PowerSet(U)) = 32
  assert 6 notin U
  test failure: complement(A) with unspecified C -> CONTEXT_REQUIRED

25. OUTPUT CONTRACT
  output: typed graph + validated relations + inference trace
  required: ontology_id, context_id, version, FLOW_HASH
  optional: provenance, confidence, evidence, validation_report
  interoperability: SetOntology, ElementOntology, EmptySetOntology,
                    SubsetOntology, PowerSetOntology, CardinalNumberOntology

END FLOW ALGEBRA