Cardinal Number Ontology

FLOW ALGEBRA
CARDINAL NUMBER ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Number Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/number-entities/cardinal-number-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-12 | Language: en-GB
Status: Mathematically grounded ontology draft; source page not independently verified.

01. FLOW IDENTITY
flow "CardinalNumberOntology" v1.0
namespace: kireth.entity_graph.mathematical_entities.number_entities
entity_type: CardinalNumber
purpose: Represent set cardinality, finite and infinite cardinal arithmetic, comparisons and inference.

02. CORE DEFINITION
Cardinal(A) := |A|, the equivalence class of sets equipotent to A (or a canonical cardinal representative).
Equipotent(A,B) <=> exists bijection f:A -> B.
|A| = |B| <=> Equipotent(A,B).
|A| <= |B| <=> exists injection f:A -> B.
|A| < |B| <=> |A| <= |B| and |A| != |B|.
Cardinality measures size without requiring an ordering of the elements.

03. ENTITY SCHEMA
CardinalNumber:
  id: URI
  notation: string
  classification: FiniteCardinal | InfiniteCardinal
  canonical_representation: optional set-theoretic representation
  defining_set: optional Set
  successor_cardinal: optional CardinalNumber
  predecessor_cardinal: optional CardinalNumber
  proof_reference: optional URI
  confidence: verified | derived | conjectural | unknown

04. FINITE CARDINALS
For every finite set A, |A| = n for a unique n in N_0.
|empty_set| = 0.
|{x}| = 1.
For finite disjoint A,B: |A union B| = |A| + |B|.
For finite A,B: |A x B| = |A| * |B|.

05. INFINITE CARDINALS
aleph_0 := |N| (countably infinite cardinal).
|Z| = aleph_0; |Q| = aleph_0.
|R| = 2^aleph_0 > aleph_0.
A set is countably infinite iff it is bijective with N.
A set is uncountable iff its cardinality is greater than aleph_0.
Cantor: |P(A)| > |A| for every set A.

06. CARDINAL SUCCESSION AND ALEPH HIERARCHY
aleph_(alpha+1) := least cardinal strictly greater than aleph_alpha.
For limit ordinal lambda: aleph_lambda := sup{aleph_alpha : alpha < lambda}.
Ordinal and cardinal are distinct types; a cardinal can be represented by an initial ordinal.
Do not identify an ordinal's order type with its cardinality.

07. CARDINAL ARITHMETIC
kappa + lambda := |A disjoint_union B| where |A|=kappa and |B|=lambda.
kappa * lambda := |A x B|.
kappa ^ lambda := |Functions(B,A)|.
kappa + 0 = kappa; kappa * 1 = kappa; kappa ^ 0 = 1 (including 0^0 = 1 in cardinal arithmetic).
For infinite cardinals kappa,lambda, under the axiom of choice:
  kappa + lambda = max(kappa,lambda).
  if kappa,lambda > 0: kappa * lambda = max(kappa,lambda).
For any cardinal kappa: 2^kappa > kappa.
Do not assume unrestricted cardinal subtraction or division.

08. SET-THEORETIC FOUNDATIONS
Foundation: ZFC unless otherwise specified.
Schroeder-Bernstein: injections A -> B and B -> A imply |A| = |B|.
Axiom of choice: every set is well-orderable; cardinals are linearly comparable.
Without choice, some infinite-cardinal arithmetic and comparability laws differ.
Continuum hypothesis (CH): 2^aleph_0 = aleph_1.
CH is independent of ZFC, assuming ZFC is consistent; never assert CH as theorem of ZFC.

09. SEMANTIC RELATIONSHIPS
CardinalNumber --measures_size_of--> Set
CardinalNumber --is_equivalent_to--> CardinalNumber [bijection witness]
CardinalNumber --is_less_than_or_equal_to--> CardinalNumber [injection witness]
CardinalNumber --is_cardinality_of--> Set
FiniteCardinal --subclass_of--> CardinalNumber
InfiniteCardinal --subclass_of--> CardinalNumber
AlephNumber --subclass_of--> InfiniteCardinal
CardinalNumber --related_to_but_distinct_from--> OrdinalNumber
PowerSet --has_cardinality--> ExponentiationCardinal

10. TYPED OPERATIONS
cardinality(Set) -> CardinalNumber
bijection(Set,Set) -> Boolean | Proof
injection(Set,Set) -> Boolean | Proof
compare(CardinalNumber,CardinalNumber) -> LT | EQ | GT | UNDETERMINED
add(CardinalNumber,CardinalNumber) -> CardinalNumber
multiply(CardinalNumber,CardinalNumber) -> CardinalNumber
exponentiate(CardinalNumber,CardinalNumber) -> CardinalNumber
power_set(Set) -> Set

11. INFERENCE RULES
R1: bijection(A,B) => cardinality(A) = cardinality(B).
R2: injection(A,B) => cardinality(A) <= cardinality(B).
R3: injection(A,B) and injection(B,A) => cardinality(A) = cardinality(B).
R4: cardinality(power_set(A)) = 2 ^ cardinality(A).
R5: cardinality(power_set(A)) > cardinality(A).
R6: finite(A) and finite(B) and disjoint(A,B) => |A union B|=|A|+|B|.
R7: |N|=aleph_0 and bijection(A,N) => countably_infinite(A).
R8: assert CH only when explicitly adopted as an additional axiom or hypothesis.

12. VALIDATION
V1: Every cardinal comparison must identify the axiomatic setting.
V2: Equality of infinite cardinalities requires a valid bijection or theorem.
V3: Finite counting must distinguish sets from multisets.
V4: A proper subset may have equal cardinality to its parent if infinite.
V5: Never infer that all infinite sets have cardinality aleph_0.
V6: Verify exponentiation direction: kappa^lambda counts functions from a lambda-sized domain to a kappa-sized codomain.
V7: Preserve explicit proof status for statements independent of ZFC.

13. EXAMPLES
|{a,b,c}| = 3.
|N| = |Z| = |Q| = aleph_0.
|N x N| = aleph_0.
|P(N)| = 2^aleph_0 = |R|.
|P(R)| = 2^(2^aleph_0) > |R|.

14. CONFIDENCE AND PROVENANCE
confidence.level: verified when a formal proof or accepted theorem supports the claim.
confidence.level: conditional when the result depends on an additional axiom.
confidence.level: unknown when neither a proof nor a supported assumption is supplied.
provenance.capture: theorem_name, assumptions, proof_reference, source_uri, revision_date.

15. SEMANTIC EQUIVALENCE
semantic_equivalence:
  "cardinality of A" <=> "size of set A" [set-theoretic context]
  "equipotent sets" <=> "sets of equal cardinality"
  "countably infinite" <=> "cardinality aleph_0"
semantic_difference:
  CardinalNumber != OrdinalNumber
  Countable != CountablyInfinite [countable can include finite sets]
  CardinalAddition != OrdinalAddition

16. FLOW HASH AND UPDATE MODEL
FLOW_HASH:
  algorithm: SHA-256
  input: canonical_utf8(flow_definition_without_hash_value)
  output: computed_by_consumer
UPDATE_MODEL:
  versioning: semantic
  update_trigger: corrected_axiom | revised_definition | new_proof | improved_mapping
  preserve: source_uri, historical_version, provenance, change_log
  reject: silent_redefinition_of_cardinal_operations

17. AI CONSUMPTION
AI_CONSUMPTION:
  mode: typed_semantic_reasoning
  required_context: axiomatic_foundation
  reasoning: definition -> witness_or_theorem -> inference -> validation
  outputs: cardinal_expression | comparison | proof_obligation | uncertainty
  safeguards:
    - Distinguish finite from infinite reasoning.
    - Do not treat CH as resolved in ZFC.
    - Do not conflate cardinality with ordering.
    - Mark dependence on the axiom of choice.

18. END FLOW
end flow "CardinalNumberOntology"