Uncountable Set Ontology

FLOW ALGEBRA
UNCOUNTABLE SET ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Set Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/set-entities/uncountable-set-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Status: Mathematically grounded specification; source page content not independently verified.

01. PURPOSE
flow "UncountableSetOntology" v1.0
  objective: represent, classify, reason about, and validate uncountable sets.
  domain: set_theory
  parent: InfiniteSetOntology
  related: [CountableSetOntology, CardinalNumberOntology, RealNumberOntology, PowerSetOntology]

02. CORE TYPES
  type Set[T]
  type NaturalNumber = N
  type Cardinal
  type Function[A,B]
  type Proposition
  entity UncountableSet[S:Set[T]]
  predicate countable(S): Proposition
  predicate uncountable(S): Proposition
  predicate injective(f): Proposition
  predicate surjective(f): Proposition
  predicate bijective(f): Proposition

03. DEFINITIONS
  countable(S) <=> exists injection f:S -> N
  uncountable(S) <=> not countable(S)
  uncountable(S) => infinite(S)
  uncountable(S) <=> no enumeration N -> S is surjective
  NOTE: The empty and finite sets are countable under this convention.

04. CARDINAL CHARACTERISATION
  cardinal(S) = |S|
  aleph0 = |N|
  uncountable(S) <=> |S| > aleph0
  continuum = |R| = 2^aleph0
  cardinal(P(S)) > cardinal(S)  # Cantor's theorem
  NOTE: Comparisons of arbitrary cardinalities rely on an appropriate set-theoretic framework; standard ZFC is assumed.

05. CANONICAL INSTANCES
  uncountable(R) = true
  uncountable((0,1)) = true
  uncountable([0,1]) = true
  uncountable(P(N)) = true
  uncountable(R\Q) = true
  countable(N) = true
  countable(Z) = true
  countable(Q) = true

06. PROOF RULES
  rule CantorDiagonal:
    input: candidate enumeration e:N -> (0,1)
    output: exists r in (0,1) not in image(e)
    consequence: uncountable((0,1))
  rule PowerSet:
    input: S:Set[T]
    output: no surjection S -> P(S)
    consequence: |P(S)| > |S|
  rule InjectedFromUncountable:
    if uncountable(A) and injective(f:A -> B)
    then uncountable(B)
  rule SurjectsOntoUncountable:
    if surjective(f:A -> B) and uncountable(B)
    then uncountable(A)
  rule Superset:
    if A subseteq B and uncountable(A)
    then uncountable(B)

07. CLOSURE AND DECOMPOSITION
  rule CountableUnion:
    if forall n in N: countable(A[n])
    then countable(union(n in N, A[n]))  # ZFC; uses countable choice in general
  rule UncountableUnionWitness:
    if uncountable(union(n in N, A[n]))
    then exists n in N: uncountable(A[n])  # Contrapositive of CountableUnion
  rule RemoveCountable:
    if uncountable(A) and countable(C)
    then uncountable(A\C)
  rule CountableImage:
    if countable(A) and f:A -> B
    then countable(image(f))
  rule UncountableImageNotAutomatic:
    if uncountable(A) and f:A -> B
    then no conclusion about uncountable(image(f)) without further conditions

08. TOPOLOGICAL AND ANALYTIC CONNECTIONS
  fact: every nonempty real interval with distinct endpoints is uncountable
  fact: every nonempty perfect subset of R is uncountable
  fact: Cantor ternary set is uncountable and has Lebesgue measure zero
  caution: uncountability does not imply positive measure
  caution: uncountability does not imply containing an interval

09. SEMANTIC ENTITY MODEL
  entity UncountableSet:
    id: uri
    label: string
    membership_predicate: optional expression
    ambient_universe: Set
    cardinality: Cardinal or symbolic
    cardinal_lower_bound: optional Cardinal
    proof_status: [proved, assumed, unknown, refuted]
    proof_method: optional string
    assumptions: list[string]
    provenance: source reference

10. KNOWLEDGE GRAPH RELATIONSHIPS
  UncountableSet --is_a--> InfiniteSet
  UncountableSet --contrasts_with--> CountableSet
  UncountableSet --has_property--> CardinalityGreaterThanAlephZero
  RealNumbers --instance_of--> UncountableSet
  PowerSetOfNaturals --instance_of--> UncountableSet
  CantorDiagonalArgument --proves--> UncountabilityOfRealInterval
  CantorTheorem --establishes--> PowerSetCardinalityIncrease

11. SEMANTIC EQUIVALENCE
  equivalent("uncountable", "not countable") under standard countability convention
  equivalent("uncountable S", "no injection S into N")
  equivalent("uncountable S", "no surjection N onto S") for nonempty S
  NOT equivalent("uncountable", "infinite")
  NOT equivalent("uncountable", "continuum-sized")
  NOT equivalent("uncountable", "non-measurable")

12. INFERENCE ENGINE
  infer Uncountable(S) from certified no-injection-to-N proof
  infer Uncountable(S) from certified injection R -> S
  infer Uncountable(S) from certified surjection S -> R
  infer Uncountable(S) from inclusion of a known uncountable subset
  prohibit inferring Uncountable(S) merely from infinite(S)
  prohibit inferring cardinal(S) = continuum merely from uncountable(S)

13. VALIDATION
  validate S is a well-defined set in selected foundational theory
  validate membership predicate and ambient universe
  validate map domains and codomains in cardinal arguments
  validate injection/surjection claims with proofs
  validate countability convention explicitly
  validate that cardinal arithmetic assumptions are stated
  validate cited theorem prerequisites before application

14. COUNTEREXAMPLES
  infinite(N) and countable(N): infinite does not imply uncountable
  uncountable(CantorSet) and measure(CantorSet)=0
  constant_map:R -> {0} has countable image despite uncountable domain
  P(N) is uncountable although N is countable

15. CONFIDENCE
  confidence: 1.0 for definitions and proved standard theorems in ZFC
  confidence: proof-dependent for user-supplied set instances
  unknown: when membership, proof, or foundational assumptions are incomplete
  never promote heuristic cardinality estimates to formal proofs

16. FLOW HASH
  canonicalise: UTF-8, stable entity identifiers, normalised symbolic expressions
  flow_hash: SHA-256(canonicalised flow payload)
  exclude volatile timestamps from semantic hash unless version policy requires them
  record hash algorithm and schema version alongside digest

17. UPDATE MODEL
  trigger: changed axioms, corrected theorem conditions, new proof, new source version
  update: preserve prior version, recompute dependent inferences, refresh provenance
  maintain: audit trail and backward-compatible identifiers where possible

18. AI CONSUMPTION
  output: typed graph triples + formal propositions + proof provenance
  expose: classification, cardinal bounds, evidence, assumptions, confidence
  require: distinction between verified theorems and hypotheses
  enable: theorem retrieval, proof checking, ontology linking, semantic search
  prohibit: fabricated proofs or unstated use of the continuum hypothesis

19. EXAMPLE EXECUTION
  input: S = P(N)
  premise: |P(N)| > |N|  [Cantor]
  premise: |N| = aleph0
  derive: |P(N)| > aleph0
  result: uncountable(P(N)) = true
  proof_status: proved

20. LIMITATIONS AND FOUNDATIONS
  foundation: ZFC unless explicitly overridden
  continuum_hypothesis: independent of ZFC; do not assert as theorem
  countable_choice: relevant to general countable-union claims outside ZFC
  distinguish proper classes from sets; do not assign ordinary set cardinalities to proper classes

END FLOW ALGEBRA