Universal Set Ontology

FLOW ALGEBRA
TITLE: Universal Set Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Set Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/universal-set-ontology-2.html
STATUS: Domain-based formalisation; source page content not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Model the universal set relative to a specified mathematical discourse, its members, subsets, complements, and well-typed set operations. A universal set here is a context-bounded universe of discourse, not an unrestricted set of all sets.

2. TYPED ENTITIES
UniverseOfDiscourse := {id: URI, context: ContextID, element_type: Type, membership_rule: Predicate, provenance: SourceRef?}
Set := {id: URI, universe: UniverseOfDiscourse, membership_rule: Predicate, cardinality: Cardinal?}
Element := {id: URI, type: Type}
Context := {id: ContextID, scope: String, assumptions: Set[Statement]}
Predicate := Element -> Boolean
Statement := {expression: String, context: ContextID, confidence: Confidence, provenance: SourceRef?}
Confidence := Real[0,1]
SourceRef := {uri: URI, retrieved_at: Timestamp?, version: String?}

3. RELATIONSHIPS
MEMBER_OF(x: Element, S: Set) := x ∈ S
SUBSET_OF(A: Set, B: Set) := A ⊆ B
PROPER_SUBSET_OF(A: Set, B: Set) := A ⊂ B
UNIVERSE_OF(S: Set, U: UniverseOfDiscourse) := S.universe = U
COMPLEMENT_OF(C: Set, A: Set, U: UniverseOfDiscourse) := C = U \ A
EQUIVALENT_UNIVERSE(U1, U2) := same_context(U1,U2) ∧ same_membership(U1,U2)

4. AXIOMS AND CONSTRAINTS
A1: ∀S, U: UNIVERSE_OF(S,U) ⇒ S ⊆ U.
A2: ∀x, U: x ∈ U iff U.membership_rule(x) is true in U.context.
A3: ∀S, U: UNIVERSE_OF(S,U) ⇒ complement_U(S) = {x ∈ U | x ∉ S}.
A4: ∀S, U: UNIVERSE_OF(S,U) ⇒ S ∪ complement_U(S) = U.
A5: ∀S, U: UNIVERSE_OF(S,U) ⇒ S ∩ complement_U(S) = ∅.
A6: ∀U: complement_U(U) = ∅ and complement_U(∅) = U.
A7: ∀A,B: operations requiring a shared universe must first verify compatible contexts and element types.
A8: ∀U: U is a defined domain of discourse; no unrestricted universal set is assumed in ZFC set theory.

5. FLOW OPERATIONS
flow DEFINE_UNIVERSE(context, element_type, membership_rule) -> UniverseOfDiscourse
  require well_formed(context, element_type, membership_rule)
  return UniverseOfDiscourse(context, element_type, membership_rule)

flow VALIDATE_SUBSET(S, U) -> Boolean
  require compatible_types(S, U) and same_context(S, U)
  return forall(x, MEMBER_OF(x,S) implies U.membership_rule(x))

flow COMPLEMENT(S, U) -> Set
  require VALIDATE_SUBSET(S,U)
  return {x in U | not MEMBER_OF(x,S)}

flow INTERSECT(A,B,U) -> Set
  require VALIDATE_SUBSET(A,U) and VALIDATE_SUBSET(B,U)
  return {x in U | MEMBER_OF(x,A) and MEMBER_OF(x,B)}

flow UNION(A,B,U) -> Set
  require VALIDATE_SUBSET(A,U) and VALIDATE_SUBSET(B,U)
  return {x in U | MEMBER_OF(x,A) or MEMBER_OF(x,B)}

6. INFERENCE RULES
R1: MEMBER_OF(x,A) and SUBSET_OF(A,U) => MEMBER_OF(x,U).
R2: SUBSET_OF(A,B) and SUBSET_OF(B,U) => SUBSET_OF(A,U).
R3: MEMBER_OF(x,U) and not MEMBER_OF(x,A) and SUBSET_OF(A,U) => MEMBER_OF(x,COMPLEMENT(A,U)).
R4: MEMBER_OF(x,A) and SUBSET_OF(A,U) => not MEMBER_OF(x,COMPLEMENT(A,U)).
R5: SUBSET_OF(A,U) => UNION(A,COMPLEMENT(A,U),U) = U.
R6: SUBSET_OF(A,U) => INTERSECT(A,COMPLEMENT(A,U),U) = ∅.
R7: SUBSET_OF(A,U) => COMPLEMENT(COMPLEMENT(A,U),U) = A.
R8: SUBSET_OF(A,B) and SUBSET_OF(B,U) => SUBSET_OF(COMPLEMENT(B,U),COMPLEMENT(A,U)).

7. EXAMPLE
U = {1,2,3,4,5}; A = {2,4}; B = {1,2,3}.
A ⊆ U; B ⊆ U.
COMPLEMENT(A,U) = {1,3,5}.
UNION(A,COMPLEMENT(A,U),U) = U.
INTERSECT(A,B,U) = {2}.

8. CONFIDENCE
CONFIDENCE(statement) := 1.0 for valid deductions under explicitly verified axioms and assumptions; otherwise record an evidence-based value and uncertainty reason. Confidence is not a substitute for proof.

9. SEMANTIC_EQUIVALENCE
SEMANTIC_EQUIVALENCE("universal set", "universe of discourse") := conditional on shared mathematical context.
SEMANTIC_EQUIVALENCE("complement of A", "U minus A") := true when the same U is fixed.
SEMANTIC_EQUIVALENCE("universal set", "set of all sets") := false in standard ZFC.

10. PROVENANCE
PROVENANCE := {source_url, source_version?, extraction_method: "domain-based draft", assumptions, author_credit}.
Every imported claim should retain its originating source and context.

11. FLOW_HASH
FLOW_HASH := SHA256(canonical_utf8(normalise({title, version, entity_schema, axioms, inference_rules, operations}))).
Recompute after substantive modifications. Hash is defined algorithmically; no uncomputed digest is asserted.

12. UPDATE_MODEL
UPDATE_MODEL(new_evidence) := validate(new_evidence) -> compare_versions -> preserve_provenance -> revise_affected_axioms_or_mappings -> rerun_consistency_checks -> recompute_FLOW_HASH -> publish_version.

13. AI_CONSUMPTION
AI_CONSUMPTION := {
  parse: typed_entities,
  scope: universe_of_discourse,
  validate: [types, context, subset_constraints, consistency],
  reason: [R1,R2,R3,R4,R5,R6,R7,R8],
  return: [conclusions, assumptions, provenance, confidence, FLOW_HASH],
  prohibit: [unscoped_complements, unrestricted_universal_set_assumption, fabricated_proofs]
}.

END FLOW ALGEBRA