Real Number System Ontology

flow "RealNumberSystemOntology" v1.0

# ============================================================
# REAL NUMBER SYSTEM ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/real-analysis/real-number-system-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Real Analysis"
ENTITY := "Real Number System"

# ---------- CORE ENTITY ----------
REAL_NUMBER_SYSTEM := {
    type: CompleteOrderedField,
    domain: RealAnalysis,
    symbol: "R",
    purpose: "Represent the complete ordered field of real numbers that provides the numerical foundation for real analysis and calculus."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    REAL_NUMBER,
    REAL_NUMBERS,
    NATURAL_NUMBERS,
    INTEGERS,
    RATIONAL_NUMBERS,
    IRRATIONAL_NUMBERS,
    FIELD,
    ORDERED_FIELD,
    COMPLETENESS,
    SUPREMUM,
    INFIMUM,
    ABSOLUTE_VALUE,
    INTERVAL,
    ARCHIMEDEAN_PROPERTY,
    DENSITY
}

# ---------- NUMBER HIERARCHY ----------
NATURAL_NUMBERS
    subset_of INTEGERS

INTEGERS
    subset_of RATIONAL_NUMBERS

RATIONAL_NUMBERS
    subset_of REAL_NUMBERS

IRRATIONAL_NUMBERS
    subset_of REAL_NUMBERS

REAL_NUMBERS
    =
RATIONAL_NUMBERS
union
IRRATIONAL_NUMBERS

RATIONAL_NUMBERS
intersection
IRRATIONAL_NUMBERS
    = EMPTY_SET

# ---------- REAL NUMBER ----------
REAL_NUMBER
    := element_of(REAL_NUMBERS)

REAL_NUMBERS
    := R

R
    -> COMPLETE_ORDERED_FIELD

# ---------- FIELD STRUCTURE ----------
FIELD_OPERATIONS := {
    ADDITION,
    SUBTRACTION,
    MULTIPLICATION,
    DIVISION_BY_NONZERO
}

R
    -> closed_under(ADDITION)
    -> closed_under(SUBTRACTION)
    -> closed_under(MULTIPLICATION)
    -> closed_under(DIVISION_BY_NONZERO)

# ---------- ADDITIVE STRUCTURE ----------
for
    a,b,c in R

a+b
    = b+a

(a+b)+c
    = a+(b+c)

exists
    0 in R

such_that

a+0
    = a

for each a
exists -a

such_that

a+(-a)
    = 0

# ---------- MULTIPLICATIVE STRUCTURE ----------
for
    a,b,c in R

a*b
    = b*a

(a*b)*c
    = a*(b*c)

exists
    1 in R

such_that

a*1
    = a

for
    a != 0

exists
    a^(-1)

such_that

a*a^(-1)
    = 1

# ---------- DISTRIBUTIVITY ----------
a*(b+c)
    =
a*b + a*c

DISTRIBUTIVE_LAW
    -> connects(ADDITION, MULTIPLICATION)

# ---------- ORDER STRUCTURE ----------
for
    a,b in R

exactly_one_of:
    a < b
    a = b
    a > b

ORDER
    -> TOTAL_ORDER

if
    a < b
then

a+c
    < b+c

if
    a < b
    AND c > 0
then

a*c
    < b*c

FIELD
    + COMPATIBLE_TOTAL_ORDER
    -> ORDERED_FIELD

# ---------- POSITIVE / NEGATIVE ----------
POSITIVE_REAL
    := x > 0

NEGATIVE_REAL
    := x < 0

NONNEGATIVE_REAL
    := x >= 0

SIGN
    -> determined_by(ORDER_RELATION_TO_ZERO)

# ---------- ABSOLUTE VALUE ----------
abs(x)
    := {
        x,  if x >= 0
        -x, if x < 0
    }

ABSOLUTE_VALUE
    -> distance_from_zero

abs(x-y)
    -> distance_between(x,y)

# ---------- ABSOLUTE VALUE PROPERTIES ----------
abs(x)
    >= 0

abs(x)
    = 0
    iff
    x = 0

abs(x*y)
    =
abs(x)*abs(y)

abs(x+y)
    <=
abs(x)+abs(y)

TRIANGLE_INEQUALITY
    -> fundamental_metric_property

# ---------- METRIC ----------
d(x,y)
    := abs(x-y)

REAL_NUMBERS
    -> METRIC_SPACE

METRIC
    -> supports(CONVERGENCE)
    -> supports(CONTINUITY)
    -> supports(OPEN_AND_CLOSED_SETS)

# ---------- COMPLETENESS ----------
LEAST_UPPER_BOUND_PROPERTY:

if
    A subset_of R
    AND A nonempty
    AND A bounded_above
then

exists
    sup(A) in R

COMPLETENESS
    -> defining_difference_between(R_AND_Q)

# ---------- SUPREMUM ----------
s
    := sup(A)

iff

s
    -> UPPER_BOUND_OF(A)

and

for every upper bound u of A:
    s <= u

SUPREMUM
    -> least_upper_bound

# ---------- INFIMUM ----------
m
    := inf(A)

iff

m
    -> LOWER_BOUND_OF(A)

and

for every lower bound l of A:
    l <= m

INFIMUM
    -> greatest_lower_bound

# ---------- CAUCHY COMPLETENESS ----------
sequence
    := (x_n)

if
    x_n is Cauchy_in_R
then

exists
    x in R

such_that

x_n -> x

CAUCHY_COMPLETENESS
    -> equivalent_to(ORDER_COMPLETENESS)
       in(REAL_NUMBER_SYSTEM)

# ---------- MONOTONE COMPLETENESS ----------
if
    sequence increasing
    AND bounded_above
then

sequence
    -> converges_in(R)

if
    sequence decreasing
    AND bounded_below
then

sequence
    -> converges_in(R)

MONOTONE_CONVERGENCE
    -> consequence_of(COMPLETENESS)

# ---------- ARCHIMEDEAN PROPERTY ----------
for
    x in R

exists
    n in NATURAL_NUMBERS

such_that

n > x

ARCHIMEDEAN_PROPERTY
    -> no_infinite_real_number

also

for
    epsilon > 0

exists
    n in NATURAL_NUMBERS

such_that

1/n < epsilon

# ---------- DENSITY OF RATIONALS ----------
for
    x,y in R
    with x < y

exists
    q in RATIONAL_NUMBERS

such_that

x < q < y

RATIONAL_NUMBERS
    -> DENSE_IN(R)

# ---------- DENSITY OF IRRATIONALS ----------
for
    x < y

exists
    r in IRRATIONAL_NUMBERS

such_that

x < r < y

IRRATIONAL_NUMBERS
    -> DENSE_IN(R)

# ---------- RATIONAL NUMBERS ----------
RATIONAL_NUMBER
    := p/q

where
    p in INTEGERS
    q in INTEGERS
    q != 0

Q
    -> ORDERED_FIELD
    -/> COMPLETE

RATIONAL_NUMBERS
    -> dense_in(R)

# ---------- IRRATIONAL NUMBERS ----------
IRRATIONAL_NUMBER
    := real_number_not_in(Q)

examples:
    sqrt(2)
    pi
    e

IRRATIONALS
    -> required_to_complete(RATIONAL_NUMBER_LINE)

# ---------- DECIMAL REPRESENTATION ----------
REAL_NUMBER
    -> may_be_represented_by(DECIMAL_EXPANSION)

RATIONAL_NUMBER
    -> terminating_or_eventually_periodic_decimal

IRRATIONAL_NUMBER
    -> nonterminating_nonperiodic_decimal

DECIMAL_REPRESENTATION
    -> not_always_unique_in_symbolic_form
       because(e.g._0.999...=1)

# ---------- INTERVALS ----------
OPEN_INTERVAL
    := (a,b)
    = {
        x in R :
        a < x < b
      }

CLOSED_INTERVAL
    := [a,b]
    = {
        x in R :
        a <= x <= b
      }

HALF_OPEN_INTERVAL
    -> [a,b)
    -> (a,b]

INTERVAL
    -> connected_subset_of(R)

# ---------- UNBOUNDED INTERVALS ----------
(a,infinity)
(-infinity,b)
[a,infinity)
(-infinity,b]

INFINITY
    -/> REAL_NUMBER

INFINITY_SYMBOL
    -> denotes(UNBOUNDED_DIRECTION)

# ---------- EXTENDED REAL NUMBERS ----------
EXTENDED_REAL_LINE
    := R union {
        -infinity,
        +infinity
    }

EXTENDED_REALS
    -/> FIELD

EXTENDED_REALS
    -> useful_for(LIMITS_AND_MEASURE_THEORY)

# ---------- CARDINALITY ----------
REAL_NUMBERS
    -> UNCOUNTABLE

RATIONAL_NUMBERS
    -> COUNTABLE

IRRATIONAL_NUMBERS
    -> UNCOUNTABLE

CANTOR_DIAGONAL_ARGUMENT
    -> proves(UNCOUNTABILITY_OF_R)

# ---------- CONTINUUM ----------
CARDINALITY(R)
    := CONTINUUM

REAL_LINE
    -> geometric_model_of(R)

POINT_ON_REAL_LINE
    <-> REAL_NUMBER

# ---------- APPLICATIONS ----------
REAL_NUMBER_SYSTEM
    -> foundation_of(REAL_ANALYSIS)
    -> foundation_of(CALCULUS)
    -> applied_to(MEASUREMENT)
    -> applied_to(GEOMETRY)
    -> applied_to(PROBABILITY)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(NUMERICAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
NATURAL_NUMBERS
    -> subset_of(INTEGERS)

INTEGERS
    -> subset_of(RATIONAL_NUMBERS)

RATIONAL_NUMBERS
    -> subset_of(REAL_NUMBERS)

REAL_NUMBERS
    -> partitioned_into(RATIONALS, IRRATIONALS)

FIELD
    + ORDER
    + COMPLETENESS
    -> REAL_NUMBER_SYSTEM

ABSOLUTE_VALUE
    -> induces(METRIC)

COMPLETENESS
    -> supports(LIMIT_AND_CONVERGENCE_THEORY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "real number system" ~= "complete ordered field of real numbers",
    "real line" ~= "geometric representation of the real numbers",
    "least upper bound property" ~= "order completeness of the real numbers",
    "absolute value" ~= "distance from zero on the real line",
    "rational number" ~= "ratio of two integers with nonzero denominator",
    "irrational number" ~= "real number that is not rational"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    real_numbers_form_complete_ordered_field: true,
    rational_numbers_are_not_complete: true,
    infinity_is_not_real_number: true,
    extended_real_line_is_not_field: true,
    rational_and_irrational_numbers_partition_reals: true,
    decimal_representation_can_have_equivalent_forms: true,
    completeness_must_not_be_confused_with_algebraic_closure: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    field_structure: 0.99,
    order_structure: 0.99,
    completeness_relation: 0.99,
    rational_irrational_relation: 0.99,
    metric_interval_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(number_or_set)
    -> classify(natural_integer_rational_irrational_or_real)
    -> identify(field_and_order_operations)
    -> analyse(bounds_if_set)
    -> compute_or_reason_about(supremum_or_infimum_if_relevant)
    -> apply(completeness_if_limit_structure_present)
    -> analyse(absolute_value_metric_or_interval_structure)
    -> apply(archimedean_or_density_properties_if_requested)
    -> distinguish(real_and_extended_real_context)
    -> validate(number_system_constraints)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed real-number-system semantic graph",
    primary_entity: REAL_NUMBER_SYSTEM,
    preserve: [
        number_class,
        field_operations,
        order_relation,
        completeness,
        bounds,
        interval_structure,
        metric_structure
    ],
    infer_through: [
        natural_numbers,
        integers,
        rationals,
        irrationals,
        suprema,
        infima,
        absolute_values,
        density,
        archimedean_property
    ],
    avoid: [
        treating_infinity_as real number,
        claiming_rationals_are complete,
        confusing_completeness_with algebraic closure,
        assuming_unique_decimal notation,
        confusing_extended_reals_with field,
        omitting_nonzero_denominator_for rational numbers
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(real_number_system_context)
    -> compare(existing_relations)
    -> preserve(field_order_and_completeness_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + REAL_NUMBER_SYSTEM
    + REAL_NUMBERS
    + FIELD_OPERATIONS
    + ORDER
    + COMPLETENESS
    + SUPREMUM
    + INFIMUM
    + ABSOLUTE_VALUE
    + ARCHIMEDEAN_PROPERTY
    + DENSITY
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> REAL_NUMBER_SYSTEM
    -> NUMBER_SET_IDENTIFICATION
    -> NUMBER_CLASSIFICATION
    -> FIELD_OPERATION_ANALYSIS
    -> ORDER_ANALYSIS
    -> BOUND_SUPREMUM_INFIMUM_ANALYSIS
    -> COMPLETENESS_ANALYSIS
    -> METRIC_INTERVAL_ANALYSIS
    -> ARCHIMEDEAN_DENSITY_ANALYSIS
    -> REAL_EXTENDED_REAL_CLASSIFICATION
    -> CONSTRAINT_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW