Infinite Set Ontology

FLOW ALGEBRA
INFINITE SET ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Set Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/set-entities/infinite-set-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13
Status: Formal mathematical specification; source webpage not independently verified.

01. PURPOSE
flow "InfiniteSetOntology" v1.0
  domain: set_theory
  entity_type: InfiniteSet
  parent: Set
  intent: classify, reason about, and validate infinite sets

02. CORE DEFINITIONS
  Set(S): collection with well-defined membership under a chosen foundation
  Finite(S): exists n in NaturalNumbers and bijection S <-> {0,...,n-1}
  Infinite(S): Set(S) AND NOT Finite(S)
  EmptySet: finite set with cardinality 0
  Infinite(S) -> S != EmptySet

03. FOUNDATIONAL CONTEXT
  foundation: ZFC unless otherwise declared
  equality: extensional equality
  universe: contextual domain; no unrestricted set of all sets
  cardinality: equivalence class under bijection, represented by a cardinal
  note: Dedekind-infinite equivalence uses a choice principle in general

04. ENTITY SCHEMA
  InfiniteSet:
    id: URI
    membership_predicate: predicate(x)
    representation: symbolic | enumerated_rule | constructive | axiomatic
    element_domain: SetOrClass
    cardinality: CardinalExpression?
    countability: countably_infinite | uncountable | unknown
    ordering: OrderStructure?
    topology: TopologicalStructure?
    proof_of_infinitude: ProofReference?
    assumptions: FoundationReference
    provenance: SourceReference
    confidence: ConfidenceRecord

05. CLASSIFICATION
  CountablyInfinite(S): exists bijection NaturalNumbers <-> S
  AtMostCountable(S): Finite(S) OR CountablyInfinite(S)
  Uncountable(S): Infinite(S) AND NOT CountablyInfinite(S)
  Infinite(S) -> CountablyInfinite(S) OR Uncountable(S) [classical logic]
  CountablyInfinite(S) -> cardinality(S) = aleph_0
  Uncountable(S) -> cardinality(S) > aleph_0 [ZFC]

06. CANONICAL EXAMPLES
  NaturalNumbers: countably_infinite
  Integers: countably_infinite
  RationalNumbers: countably_infinite
  RealNumbers: uncountable
  Interval(0,1): uncountable
  PowerSet(NaturalNumbers): uncountable
  EvenNaturalNumbers: countably_infinite

07. INFINITUDE CRITERIA
  InjectiveMap(NaturalNumbers,S) -> Infinite(S)
  Bijection(S,ProperSubset(S)) -> DedekindInfinite(S)
  DedekindInfinite(S) -> Infinite(S)
  Infinite(S) -> DedekindInfinite(S) [ZFC; not valid in all ZF models]
  UnboundedSubsetOfNaturalNumbers(S) -> Infinite(S)
  FiniteUnionOfFiniteSets -> FiniteSet

08. CARDINALITY RELATIONSHIPS
  Bijection(A,B) -> cardinality(A) = cardinality(B)
  Injection(A,B) -> cardinality(A) <= cardinality(B) [ZFC]
  Injection(A,B) AND Injection(B,A) -> Bijection(A,B) [Cantor-Bernstein]
  Set(S) -> cardinality(S) < cardinality(PowerSet(S)) [Cantor]
  cardinality(NaturalNumbers) = aleph_0
  cardinality(RealNumbers) = 2^aleph_0
  cardinality(PowerSet(NaturalNumbers)) = 2^aleph_0
  continuum_hypothesis: independent_of_ZFC_if_ZFC_consistent

09. SUBSET RULES
  Infinite(S) AND Finite(F) AND F subset S -> Infinite(S \ F)
  Infinite(S) AND S subset T -> Infinite(T)
  Infinite(S) -> exists countably_infinite subset of S [ZFC]
  Infinite(S) does NOT imply every proper subset is infinite
  ProperSubset(T,S) does NOT imply cardinality(T) < cardinality(S)

10. UNION AND INTERSECTION
  Infinite(A) OR Infinite(B) -> Infinite(A union B)
  CountablyInfinite(A) AND CountablyInfinite(B) -> CountablyInfinite(A union B)
  Infinite(A) AND Infinite(B) does NOT imply Infinite(A intersect B)
  IntersectionOfInfiniteSets may be empty, finite, or infinite
  CountableUnionOfCountableSets -> AtMostCountable [ZFC]

11. CARTESIAN PRODUCTS
  CountablyInfinite(A) AND CountablyInfinite(B) -> CountablyInfinite(A x B)
  Infinite(A) AND Nonempty(B) -> Infinite(A x B)
  cardinality(NaturalNumbers x NaturalNumbers) = aleph_0
  cardinality(RealNumbers x RealNumbers) = cardinality(RealNumbers)

12. POWER SETS
  Infinite(S) -> Infinite(PowerSet(S))
  cardinality(PowerSet(S)) > cardinality(S)
  PowerSet(NaturalNumbers): uncountable
  FiniteSubsets(NaturalNumbers): countably_infinite

13. ORDER AND TOPOLOGY
  InfiniteSet does not entail a particular ordering or topology
  NaturalNumbers with usual order: well_ordered
  Integers with usual order: not well_ordered
  RealNumbers with usual order: densely_ordered
  InfiniteBoundedSubset(RealNumbers) -> has accumulation point [Bolzano-Weierstrass]
  InfiniteSubset(RealNumbers) need not be bounded

14. SEMANTIC RELATIONSHIPS
  InfiniteSet IS_A Set
  InfiniteSet DISJOINT_WITH FiniteSet
  CountablyInfiniteSet IS_A InfiniteSet
  UncountableSet IS_A InfiniteSet
  InfiniteSet HAS_CARDINALITY CardinalNumber
  InfiniteSet MAY_HAVE Ordering
  InfiniteSet MAY_HAVE Topology
  InfiniteSet MAY_BE_SUBSET_OF InfiniteSet
  InfiniteSet CONNECTS_TO CardinalNumberOntology
  InfiniteSet CONNECTS_TO FiniteSetOntology
  InfiniteSet CONNECTS_TO CountableSetOntology
  InfiniteSet CONNECTS_TO PowerSetOntology

15. KNOWLEDGE GRAPH TRIPLES
  (InfiniteSet, subClassOf, Set)
  (InfiniteSet, disjointWith, FiniteSet)
  (CountablyInfiniteSet, subClassOf, InfiniteSet)
  (UncountableSet, subClassOf, InfiniteSet)
  (NaturalNumbers, instanceOf, CountablyInfiniteSet)
  (RealNumbers, instanceOf, UncountableSet)
  (PowerSetNaturalNumbers, hasCardinality, Continuum)

16. INFERENCE ENGINE
  rule R1: certified_injection(NaturalNumbers,S) => Infinite(S)
  rule R2: certified_bijection(NaturalNumbers,S) => CountablyInfinite(S)
  rule R3: certified_injection(RealNumbers,S) => Uncountable(S)
  rule R4: Infinite(S) AND S subset T => Infinite(T)
  rule R5: Infinite(S) AND Finite(F) => Infinite(S \ F)
  rule R6: Set(S) => cardinality(PowerSet(S)) > cardinality(S)
  guard: distinguish proven, conjectured, and unknown classifications

17. SEMANTIC EQUIVALENCE
  same_members: A = B iff forall x (x in A iff x in B)
  equipotent: A ~ B iff exists bijection A <-> B
  A ~ B does NOT imply A = B
  infinite_set NOT_EQ infinity_as_extended_real_symbol
  cardinal_number NOT_EQ ordinal_number in general

18. VALIDATION
  require well_defined_membership_or_foundational_construction
  reject infinite_and_finite_for_same_set_under_same_assumptions
  require explicit foundation for choice-sensitive inferences
  verify claimed bijections and injections
  never infer uncountability solely from density
  never infer finiteness from boundedness
  never infer countability from enumerated examples alone
  report undecidable_or_unknown where applicable

19. CONFIDENCE
  CONFIDENCE:
    theorem_with_formal_proof: 1.00
    validated_derivation: 0.99
    supported_mathematical_reference: 0.95
    unverified_claim: unknown
  confidence is evidential metadata, not truth value

20. FLOW HASH
  FLOW_HASH:
    method: SHA-256
    input: canonical_utf8(ontology_id,version,axioms,rules,dependencies)
    hash_value: compute_at_publication
    note: no fabricated digest

21. UPDATE MODEL
  UPDATE_MODEL:
    version_policy: semantic_versioning
    triggers: corrected_axiom | new_proof | changed_dependency | revised_foundation
    preserve_prior_versions: true
    invalidate_affected_inferences: true
    recompute_flow_hash: true

22. AI CONSUMPTION
  AI_CONSUMPTION:
    parse_entities: true
    typed_relationships: true
    reasoning_mode: proof_aware
    distinguish_cardinality_from_membership: true
    preserve_assumptions: true
    output: classification | derivation | counterexample | uncertainty
    hallucination_guard: never assert unsupported cardinal equality

23. TEST CASES
  test NaturalNumbers -> countably_infinite: PASS
  test Integers -> countably_infinite: PASS
  test RationalNumbers -> countably_infinite: PASS
  test RealNumbers -> uncountable: PASS
  test EvenNaturalNumbers -> countably_infinite: PASS
  test EmptySet -> infinite: FAIL
  test intersection_of_any_two_infinite_sets -> infinite: FAIL
  test every_infinite_set_has_countably_infinite_subset: PASS under ZFC
  test all_infinite_sets_same_cardinality: FAIL

24. PROVENANCE AND AUTHORSHIP
  project: Kireth Semantic Library
  author: Mark Higgins
  co_author: Sky (Recursive AI)
  source_page: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/set-entities/infinite-set-ontology-2.html
  source_verification: not independently verified
  specification_basis: established set theory
  publication_note: validate against the source page before claiming page-level alignment

END FLOW ALGEBRA