Real Number Ontology

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

01. ONTOLOGY IDENTITY
ontology_id: kireth:number:real
canonical_entity: RealNumber
symbol: ℝ
entity_type: MathematicalEntity > NumberEntity > RealNumber
scope: Complete ordered field of real numbers, including rational and irrational numbers.

02. FORMAL DEFINITION
RealNumber(x) := x ∈ ℝ
ℝ := the unique complete Archimedean ordered field up to order-preserving field isomorphism.
ℚ ⊂ ℝ; ℝ = ℚ ∪ (ℝ \ ℚ); ℚ ∩ (ℝ \ ℚ) = ∅.
For all x ∈ ℝ: x is rational XOR irrational.
Real numbers can be constructed as Dedekind cuts of ℚ or equivalence classes of Cauchy sequences in ℚ.

03. TYPED ENTITIES
RealNumber: value in ℝ
RationalNumber: value in ℚ
IrrationalNumber: value in ℝ \ ℚ
Integer: value in ℤ
NaturalNumber: value in ℕ (convention for zero explicitly declared)
Interval: subset of ℝ with specified endpoint inclusion
RealSequence: mapping ℕ → ℝ
RealFunction: mapping D ⊆ ℝ → ℝ
Bound: lower or upper bound of a subset of ℝ
Limit: convergence target in ℝ, when it exists

04. CORE RELATIONSHIPS
SUBCLASS_OF(RationalNumber, RealNumber)
SUBCLASS_OF(IrrationalNumber, RealNumber)
SUBCLASS_OF(Integer, RationalNumber)
SUBCLASS_OF(NaturalNumber, Integer)
DISJOINT(RationalNumber, IrrationalNumber)
PARTITION(RealNumber, RationalNumber, IrrationalNumber)
ORDERED_BY(RealNumber, ≤)
SUPPORTS_OPERATION(RealNumber, addition, subtraction, multiplication, division_nonzero)
SUPPORTS_PROPERTY(RealNumber, least_upper_bound)

05. FIELD OPERATIONS
∀a,b ∈ ℝ: a+b ∈ ℝ; a−b ∈ ℝ; ab ∈ ℝ.
∀a,b ∈ ℝ, b≠0: a/b ∈ ℝ.
Associativity: (a+b)+c=a+(b+c); (ab)c=a(bc).
Commutativity: a+b=b+a; ab=ba.
Distributivity: a(b+c)=ab+ac.
Identities: a+0=a; a·1=a.
Inverses: a+(−a)=0; a·a⁻¹=1 for a≠0.
0≠1.

06. ORDER AXIOMS
Exactly one of a<b, a=b, a>b holds.
If a≤b and b≤c then a≤c.
If a≤b then a+c≤b+c.
If a≤b and 0≤c then ac≤bc.
For a≠0: a²>0.
Absolute value: |a|=max(a,−a).
Distance: d(a,b)=|a−b|.

07. COMPLETENESS AND TOPOLOGY
Least-upper-bound property: every nonempty S⊆ℝ bounded above has sup(S)∈ℝ.
Greatest-lower-bound property: every nonempty S⊆ℝ bounded below has inf(S)∈ℝ.
Every Cauchy sequence in ℝ converges in ℝ.
Every bounded sequence in ℝ has a convergent subsequence.
Every closed bounded interval [a,b] is compact.
The rationals and irrationals are each dense in ℝ.
ℝ is uncountable; ℚ is countable.

08. REPRESENTATION
Exact: integer, fraction, algebraic expression, symbolic constant, Dedekind cut, Cauchy sequence.
Approximate: decimal expansion, floating-point value, bounded interval.
A real has a decimal expansion, which may be nonterminating and nonrepeating.
Terminating or eventually repeating decimal expansions denote rational numbers.
Non-eventually-repeating decimal expansions denote irrational numbers.
Decimal representations are not always unique (e.g. 0.999…=1).
Binary floating-point is generally an approximation, not an exact representation of arbitrary reals.

09. EXAMPLES AND NON-EXAMPLES
EXAMPLE(0, RealNumber)
EXAMPLE(−7, RealNumber)
EXAMPLE(3/4, RealNumber)
EXAMPLE(√2, RealNumber)
EXAMPLE(π, RealNumber)
EXAMPLE(e, RealNumber)
NOT_EXAMPLE(i, RealNumber)
NOT_EXAMPLE(+∞, RealNumber)
NOT_EXAMPLE(−∞, RealNumber)
NOTE: ±∞ belong to the extended real number system, not ℝ.

10. INFERENCE RULES
IF Integer(x) THEN RationalNumber(x) AND RealNumber(x).
IF RationalNumber(x) THEN RealNumber(x).
IF IrrationalNumber(x) THEN RealNumber(x) AND NOT RationalNumber(x).
IF RealNumber(x) AND NOT RationalNumber(x) THEN IrrationalNumber(x).
IF RealNumber(a) AND RealNumber(b) THEN RealNumber(a+b) AND RealNumber(ab).
IF RealNumber(a) AND RealNumber(b) AND b≠0 THEN RealNumber(a/b).
IF RealNumber(a) AND a≥0 THEN RealNumber(√a).
IF RealNumber(a) AND a<0 THEN NOT RealNumber(√a) under the principal complex square-root interpretation.
IF RationalNumber(a) AND IrrationalNumber(b) THEN IrrationalNumber(a+b).
IF RationalNumber(a) AND a≠0 AND IrrationalNumber(b) THEN IrrationalNumber(ab).
Do not infer that the sum or product of two irrational numbers is irrational.

11. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE(RealNumber, 'real number', context=mathematics)
SYMBOL_EQUIVALENCE(RealNumber, ℝ)
NOT_EQUIVALENT(RealNumber, ComplexNumber)
NOT_EQUIVALENT(RealNumber, ExtendedRealNumber)
IDENTITY_RULE: 1/2 = 0.5 = 2/4 as exact real values.
CONTEXT_RULE: lexical word 'real' does not automatically indicate RealNumber.

12. VALIDATION CONSTRAINTS
CHECK: division denominator ≠ 0.
CHECK: comparison operators require ordered-compatible operands.
CHECK: distinguish exact values from approximations.
CHECK: NaN is not a real number.
CHECK: floating-point ±Infinity is not an element of ℝ.
CHECK: √(negative real) is not real in ordinary real arithmetic.
CHECK: every inferred irrational classification has a valid proof or trusted theorem.
CHECK: do not assert decimal truncations are exact equalities without proof.

13. CONFIDENCE AND PROVENANCE
confidence.kind: proof | exact_definition | certified_computation | numerical_estimate
confidence.rule: mathematical theorem supported by valid proof → exact
confidence.rule: floating-point estimate alone → approximate
provenance.fields: source, theorem_or_rule, derivation, precision, timestamp, version
uncertainty: retain interval/error bounds when approximate.

14. FLOW EXECUTION
INPUT: mathematical expression or number entity
PARSE → NORMALISE → TYPE_CHECK → CLASSIFY → VALIDATE
→ APPLY_FIELD_RULES → APPLY_ORDER_RULES → INFER_RELATIONSHIPS
→ ATTACH_PROVENANCE → SERIALISE → AI_CONSUMPTION
On type or domain error: emit explicit validation failure, not fabricated real output.

15. UPDATE MODEL
UPDATE_MODEL.version: 1.0
UPDATE_MODEL.compatibility: preserve ℝ identity and field axioms
UPDATE_MODEL.trigger: new formalisation, corrected relation, validated extension
UPDATE_MODEL.policy: record revisions; invalidate dependent claims when premises change
FLOW_HASH: compute SHA-256 over canonical UTF-8 normalised flow content at publication time; no uncomputed digest is asserted here.

16. AI CONSUMPTION
AI_CONSUMPTION.entity: RealNumber
AI_CONSUMPTION.symbol: ℝ
AI_CONSUMPTION.primary_use: mathematical classification, reasoning, symbolic operations and semantic retrieval
AI_CONSUMPTION.requires: domain checking, exact/approximate distinction, proof-aware inference
AI_CONSUMPTION.outputs: typed facts, relationships, proofs or derivations, error bounds, provenance
AI_CONSUMPTION.safety: reject division by zero, unsupported irrationality claims and false precision

END FLOW ALGEBRA