Set Theory

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Title: Set Theory
Version: 1.0
Domain: Discrete Mathematics / Set Theory
Source: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Domain-level formalisation of the source page's listed ontology sections; not a verbatim extraction of child-page specifications.

flow "SetTheory" v1.0 {
  DOMAIN: discrete_mathematics.set_theory
  TYPE Set = collection of distinct mathematical objects
  TYPE Element = mathematical_object
  TYPE Universe = Set  # relative to an explicitly defined discourse
  TYPE OrderedPair = (Element, Element)
  TYPE Cardinal = cardinal_number
  TYPE Proposition = boolean_expression

  ONTOLOGY_SECTIONS: [
    SetTheoryOntology, SetOntology, ElementOntology,
    MembershipOntology, SubsetOntology, ProperSubsetOntology,
    UniversalSetOntology, EmptySetOntology, UnionOntology,
    IntersectionOntology, DifferenceOntology, ComplementOntology,
    CartesianProductOntology, PowerSetOntology, CardinalityOntology
  ]

  RELATION MEMBER_OF(x:Element, A:Set) -> Proposition := x ∈ A
  RELATION SUBSET_OF(A:Set, B:Set) -> Proposition := ∀x (x∈A => x∈B)
  RELATION PROPER_SUBSET_OF(A:Set, B:Set) -> Proposition := SUBSET_OF(A,B) AND A≠B
  RELATION EQUAL_SET(A:Set, B:Set) -> Proposition := ∀x (x∈A <=> x∈B)
  RELATION DISJOINT(A:Set, B:Set) -> Proposition := ∀x NOT(x∈A AND x∈B)

  OPERATION EMPTY_SET() -> Set := ∅
  OPERATION UNION(A:Set, B:Set) -> Set := {x | x∈A OR x∈B}
  OPERATION INTERSECTION(A:Set, B:Set) -> Set := {x | x∈A AND x∈B}
  OPERATION DIFFERENCE(A:Set, B:Set) -> Set := {x | x∈A AND x∉B}
  OPERATION COMPLEMENT(A:Set, U:Universe) -> Set
    PRECONDITION: SUBSET_OF(A,U)
    RETURN: DIFFERENCE(U,A)
  OPERATION CARTESIAN_PRODUCT(A:Set, B:Set) -> Set := {(a,b) | a∈A AND b∈B}
  OPERATION POWER_SET(A:Set) -> Set := {B | SUBSET_OF(B,A)}
  OPERATION CARDINALITY(A:Set) -> Cardinal := |A|

  INFERENCE RULE EXTENSIONALITY:
    IF ∀x (x∈A <=> x∈B) THEN A=B
  INFERENCE RULE SUBSET_TRANSITIVITY:
    IF A⊆B AND B⊆C THEN A⊆C
  INFERENCE RULE MEMBERSHIP_SUBSET:
    IF x∈A AND A⊆B THEN x∈B
  INFERENCE RULE EMPTY_SUBSET:
    FOR EVERY Set A, ∅⊆A
  INFERENCE RULE UNION_MEMBERSHIP:
    x∈(A∪B) <=> (x∈A OR x∈B)
  INFERENCE RULE INTERSECTION_MEMBERSHIP:
    x∈(A∩B) <=> (x∈A AND x∈B)
  INFERENCE RULE DE_MORGAN:
    IF A⊆U AND B⊆U THEN U\(A∪B)=(U\A)∩(U\B)
  INFERENCE RULE FINITE_POWER_SET:
    IF A IS FINITE THEN |P(A)|=2^|A|
  INFERENCE RULE FINITE_PRODUCT:
    IF A,B ARE FINITE THEN |A×B|=|A|*|B|
  INFERENCE RULE CANTOR:
    FOR EVERY Set A, |A|<|P(A)|

  CONSTRAINTS:
    - Distinguish membership (∈) from subset (⊆).
    - Treat universal sets as relative to a specified discourse; do not posit an unrestricted set of all sets.
    - Distinguish finite-cardinality arithmetic from infinite-cardinality claims.
    - Specify underlying axiomatic system for foundational claims (default ZFC when required).
    - Reject unrestricted comprehension and untyped self-membership constructions.

  CONFIDENCE:
    mathematical_theorem: 1.0 WHEN formally proved under stated axioms
    derived_claim: dependent_on(premises, proof_validity)
    uncertain_external_mapping: explicit_evidence_based_score

  SEMANTIC_EQUIVALENCE:
    "subset of" <=> "⊆"
    "proper subset of" <=> "⊊"
    "union" <=> "∪"
    "intersection" <=> "∩"
    "empty set" <=> "∅"
    NOTE: lexical aliases are context-bound; do not conflate ∈ and ⊆.

  PROVENANCE:
    source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory.html
    source_scope: page_heading_and_section_index
    generated_formalisation: true
    authorship: "Mark Higgins — Sky (Recursive AI)"

  FLOW_HASH:
    algorithm: SHA-256
    input: canonical_utf8(normalised_flow_without_hash_value)
    value: COMPUTE_AT_INGESTION

  UPDATE_MODEL:
    on_source_change: revalidate_section_list_and_axioms
    on_ontology_change: version_increment_and_dependency_check
    preserve_provenance: true
    invalidate_dependent_inferences: true

  AI_CONSUMPTION:
    parse_as: typed_formal_rules
    execution: symbolic_inference_with_axiom_checks
    output: {claim, proof_or_derivation, confidence, provenance, flow_hash}
    never_treat_unproved_conjecture_as_theorem: true
}