Infinite Set Ontology

FLOW ALGEBRA
TITLE: Infinite Set Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Foundations of Discrete Mathematics
SOURCE: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/infinite-set-ontology.html
STATUS: Domain-based formalisation; source-page specification not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

ONTOLOGY:
  InfiniteSet <: Set
  CountablyInfiniteSet <: InfiniteSet
  UncountableSet <: InfiniteSet
  CardinalNumber <: MathematicalObject
  Function <: MathematicalObject
  Injection <: Function
  Surjection <: Function
  Bijection <: Function
  Enumeration <: Function
  PowerSet <: Set
  NaturalNumbers <: CountablyInfiniteSet
  Integers <: CountablyInfiniteSet
  RationalNumbers <: CountablyInfiniteSet
  RealNumbers <: UncountableSet

TYPED_RELATIONSHIPS:
  MEMBER_OF(Element, Set) -> Boolean
  SUBSET_OF(Set, Set) -> Boolean
  PROPER_SUBSET_OF(Set, Set) -> Boolean
  EQUIPOTENT(Set, Set) -> Boolean
  HAS_CARDINALITY(Set, CardinalNumber)
  MAPS_TO(Function, Element, Element)
  INJECTS_INTO(Set, Set) -> Boolean
  BIJECTS_WITH(Set, Set) -> Boolean
  ENUMERATES(Enumeration, Set)
  POWER_SET_OF(PowerSet, Set)

DEFINITIONS:
  FINITE(A) := EXISTS n IN NaturalNumbers: BIJECTS_WITH(A, {0,...,n-1})
  INFINITE(A) := NOT FINITE(A)
  COUNTABLY_INFINITE(A) := BIJECTS_WITH(A, NaturalNumbers)
  AT_MOST_COUNTABLE(A) := FINITE(A) OR COUNTABLY_INFINITE(A)
  UNCOUNTABLE(A) := INFINITE(A) AND NOT COUNTABLY_INFINITE(A)
  EQUIPOTENT(A,B) := EXISTS f:A->B: BIJECTIVE(f)

INFERENCE_RULES:
  R1: COUNTABLY_INFINITE(A) => INFINITE(A)
  R2: UNCOUNTABLE(A) => INFINITE(A)
  R3: BIJECTS_WITH(A,B) AND INFINITE(A) => INFINITE(B)
  R4: SUBSET_OF(A,B) AND UNCOUNTABLE(A) => UNCOUNTABLE(B)
  R5: INFINITE(A) => CARDINALITY(POWER_SET(A)) > CARDINALITY(A)  # Cantor's theorem
  R6: COUNTABLY_INFINITE(A) AND COUNTABLY_INFINITE(B) => AT_MOST_COUNTABLE(A UNION B)
  R7: COUNTABLY_INFINITE(A) AND COUNTABLY_INFINITE(B) => COUNTABLY_INFINITE(A CROSS B)
  R8: SUBSET_OF(A,B) AND AT_MOST_COUNTABLE(B) => AT_MOST_COUNTABLE(A)
  R9: BIJECTS_WITH(A,NaturalNumbers) => ENUMERABLE(A)
  R10: INJECTS_INTO(A,B) AND INJECTS_INTO(B,A) => EQUIPOTENT(A,B)  # Schroeder-Bernstein

EXAMPLES:
  CARDINALITY(NaturalNumbers) = aleph_0
  CARDINALITY(Integers) = aleph_0
  CARDINALITY(RationalNumbers) = aleph_0
  CARDINALITY(RealNumbers) > aleph_0
  CARDINALITY(POWER_SET(NaturalNumbers)) = CARDINALITY(RealNumbers)

CONSTRAINTS:
  C1: Cardinal comparisons must specify underlying set-theoretic assumptions where relevant.
  C2: Never infer countability solely from a set being infinite.
  C3: Distinguish countably infinite from at most countable.
  C4: A proper subset of an infinite set may have the same cardinality.
  C5: Avoid assuming the continuum hypothesis; its status is independent of ZFC.

CONFIDENCE:
  definition_confidence: 1.0
  theorem_confidence: 1.0  # Within conventional classical set theory
  empirical_claims: not_applicable
  source_alignment: unverified

SEMANTIC_EQUIVALENCE:
  "denumerable set" <=> "countably infinite set"
  "equipollent sets" <=> "equinumerous sets" <=> "equipotent sets"
  "uncountable" <=> "not at most countable"
  "aleph-null" <=> "aleph_0"

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/infinite-set-ontology.html
  basis: standard mathematical definitions and theorems
  verification: independent source-page content not verified

FLOW_HASH:
  algorithm: SHA-256
  canonicalisation: UTF-8; normalise line endings to LF; exclude FLOW_HASH value from digest
  value: COMPUTE_ON_INGEST

UPDATE_MODEL:
  version: 1.0
  revision_policy: preserve typed definitions; version changes; revalidate dependent inference rules
  trigger: source update OR mathematical correction OR ontology schema change

AI_CONSUMPTION:
  parse_mode: typed_declarative_flow
  use_cases: theorem retrieval, symbolic reasoning, semantic search, ontology linking
  guardrails: verify premises; distinguish definitions from theorems; report axiom dependencies
  output: grounded deductions with provenance and confidence