Infinite Number Ontology

FLOW ALGEBRA
INFINITE NUMBER ONTOLOGY
Kireth Semantic Library | Mathematical Entities / Number Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/number-entities/infinite-number-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-12 | Locale: en-GB

01. ONTOLOGY IDENTITY
flow "InfiniteNumberOntology" v1.0
ontology_id: kireth.number.infinite
entity_type: MathematicalConcept
canonical_label: Infinite Number
scope: infinite cardinalities, infinite ordinals, and extended-number symbols
purpose: distinguish rigorously different uses of infinity in mathematical reasoning

02. TERMINOLOGICAL SAFEGUARD
rule: Infinity is not an ordinary real or complex number.
rule: Infinite cardinals and ordinals are mathematical objects in set theory.
rule: +infinity and -infinity in the extended real line are added symbols, not finite real numbers.
rule: A limit tending to infinity does not assert convergence to a real number.
rule: No largest natural number exists.

03. ENTITY SCHEMA
entity InfiniteConcept {
  id: string
  label: string
  framework: enum[set_theory, extended_real, analysis, projective_geometry, other]
  kind: enum[cardinal, ordinal, extended_symbol, limiting_behaviour]
  notation: string
  definition: string
  assumptions: list[string]
  source_reference: string?
  confidence: decimal[0,1]
}

04. CORE SET-THEORETIC ENTITIES
entity AlephZero: cardinality(N) = aleph_0
entity Omega: least infinite ordinal = omega
entity ContinuumCardinality: cardinality(R) = 2^(aleph_0)
relationship: cardinality(omega) = aleph_0
relationship: aleph_0 < 2^(aleph_0)  # Cantor's theorem
constraint: The continuum hypothesis is independent of ZFC (assuming ZFC is consistent).

05. FINITE VERSUS INFINITE SETS
predicate finite(A) := A is bijective with {0,...,n-1} for some n in N
predicate infinite(A) := not finite(A)
predicate countably_infinite(A) := exists bijection f: N -> A
predicate uncountable(A) := infinite(A) and not countably_infinite(A)
rule: countably_infinite(N) = true
rule: countably_infinite(Z) = true
rule: countably_infinite(Q) = true
rule: uncountable(R) = true
rule: every subset of N is finite or countably infinite

06. CARDINAL ARITHMETIC
context: ZFC; kappa and lambda are infinite cardinals
rule: kappa + lambda = max(kappa,lambda)
rule: kappa * lambda = max(kappa,lambda)
rule: kappa < 2^kappa
rule: cardinality(P(A)) = 2^cardinality(A)
constraint: Cardinal exponentiation is not generally reducible to a maximum.

07. ORDINAL STRUCTURE
predicate successor_ordinal(alpha+1)
predicate limit_ordinal(lambda) := lambda != 0 and lambda has no immediate predecessor
rule: omega is a limit ordinal
rule: 1 + omega = omega
rule: omega + 1 > omega
rule: ordinal addition is not commutative
rule: ordinal multiplication is not commutative
constraint: Do not interchange ordinal and cardinal arithmetic.

08. EXTENDED REAL NUMBERS
structure: R_extended = R union {-infinity,+infinity}
order: -infinity < x < +infinity for all x in R
rule: x + (+infinity) = +infinity for finite real x
rule: x + (-infinity) = -infinity for finite real x
rule: (+infinity) + (-infinity) is undefined
rule: 0 * (+infinity) is undefined in standard extended-real arithmetic
rule: (+infinity)/(+infinity) is undefined
rule: infinity - infinity is an indeterminate form in limits
constraint: Extended-real operations are partial; do not treat R_extended as a field.

09. LIMIT SEMANTICS
predicate tends_to_positive_infinity(f,x->a) := for every M in R, eventually f(x)>M
predicate tends_to_negative_infinity(f,x->a) := for every M in R, eventually f(x)<M
predicate sequence_diverges_to_positive_infinity(a_n) := for every M in R, exists N, for all n>=N: a_n>M
rule: lim(n->infinity) n = +infinity in extended-real notation
rule: lim(n->infinity) (1/n) = 0
constraint: An unbounded sequence need not tend to +infinity (e.g. (-1)^n*n).

10. ORDER AND COMPARISON
rule: all natural numbers are finite
rule: for every n in N, exists m in N with m>n
rule: aleph_0 is the least infinite cardinal
rule: omega is the least infinite ordinal
constraint: Do not compare an extended-real infinity with aleph_0 using ordinary numeric order.

11. SEMANTIC RELATIONSHIPS
InfiniteNumber --broader_context--> MathematicalEntity
AlephZero --instance_of--> InfiniteCardinal
Omega --instance_of--> InfiniteOrdinal
PositiveInfinity --instance_of--> ExtendedRealSymbol
NegativeInfinity --instance_of--> ExtendedRealSymbol
CountablyInfinite --property_of--> Set
Uncountable --property_of--> Set
InfiniteLimit --describes--> LimitingBehaviour
InfiniteCardinal --distinct_from--> ExtendedRealSymbol
InfiniteOrdinal --distinct_from--> InfiniteCardinal

12. SEMANTIC EQUIVALENCE
alias: aleph-null <=> aleph_0
alias: first infinite ordinal <=> omega
alias: positive infinity symbol <=> +infinity (within extended-real context)
constraint: omega != aleph_0 as typed objects, although cardinality(omega)=aleph_0.
constraint: Do not merge all appearances of 'infinity' into one numerical entity.

13. REASONING RULES
infer countably_infinite(A) if a verified bijection N <-> A exists
infer equinumerous(A,B) if a verified bijection A <-> B exists
infer infinite(A) if A has a countably infinite subset
infer uncountable(P(N)) by Cantor's theorem
infer undefined_expression if extended-real expression is +infinity + -infinity
infer ordinal_noncommutativity from 1+omega=omega and omega+1>omega
constraint: Proof obligations and ambient axioms must accompany advanced set-theoretic claims.

14. KNOWLEDGE GRAPH REPRESENTATION
node: InfiniteNumberOntology {type: Ontology}
node: AlephZero {type: InfiniteCardinal, notation: aleph_0}
node: Omega {type: InfiniteOrdinal, notation: omega}
node: PositiveInfinity {type: ExtendedRealSymbol, notation: +infinity}
edge: Omega --has_cardinality--> AlephZero
edge: PositiveInfinity --belongs_to--> ExtendedRealLine
edge: AlephZero --cardinality_of--> NaturalNumbers
edge: ContinuumCardinality --cardinality_of--> RealNumbers

15. VALIDATION
check: distinguish finite real, extended real, cardinal, ordinal, and limit contexts
check: reject division by infinity as an ordinary real operation
check: reject treating infinity as the greatest natural number
check: preserve operand types in arithmetic expressions
check: verify cited theorem hypotheses and axiomatic framework
check: use exact symbolic notation when possible

16. CONFIDENCE
confidence.method: theorem_or_definition_grounded
confidence.level: 1.0 for standard definitions and proved theorems under stated assumptions
confidence.level: lower for ambiguous natural-language references until disambiguated
confidence.rule: confidence is evidence-based, not a substitute for mathematical proof

17. FLOW HASH AND PROVENANCE
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonicalised ontology text excluding hash value
FLOW_HASH.value: compute_on_publication
provenance.source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/number-entities/infinite-number-ontology.html
provenance.status: mathematical synthesis; webpage content not independently verified
provenance.authors: Mark Higgins — Sky (Recursive AI)

18. UPDATE MODEL
UPDATE_MODEL.trigger: correction, new formalisation, revised cross-ontology mapping
UPDATE_MODEL.versioning: semantic versioning
UPDATE_MODEL.requirements: preserve definitions, record changed axioms, rerun validation
UPDATE_MODEL.backward_compatibility: retain stable identifiers or publish mappings

19. AI CONSUMPTION
AI_CONSUMPTION.mode: typed symbolic retrieval and theorem-aware reasoning
AI_CONSUMPTION.entity_resolution: context_sensitive
AI_CONSUMPTION.must_disambiguate: infinity, infinite number, omega, aleph_0
AI_CONSUMPTION.output: entity, type, definition, assumptions, relations, confidence, provenance
AI_CONSUMPTION.guardrail: never infer ordinary field arithmetic for infinite symbols

20. EXAMPLES
example: cardinality(N)=aleph_0
example: cardinality(R)>aleph_0
example: omega+1 != 1+omega
example: lim(x->+infinity) 1/x = 0
example: +infinity - +infinity is not assigned a numeric value

END FLOW ALGEBRA