CardinalNumberΒΆ

card.spad line 1

Members of the domain CardinalNumber are values indicating the cardinality of sets, both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. If x = \#X and y = \#Y then x+y = \#(X+Y) disjoint union x-y = \#(X-Y) relative complement x*y = \#(X*Y) cartesian product x^y = \#(X^Y) X^Y = \{g| g: Y->X\} The non-negative integers have a natural construction as cardinals 0 = \#\{\}, 1 = \{0\}, 2 = \{0, 1\}, ..., n = \{i| 0 <= i < n\}. That 0 acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. The generalized continuum hypothesis asserts center{2^Aleph i = Aleph(i+1)} and is independent of the axioms of set theory [Goedel 1940]. Three commonly encountered cardinal numbers are a = \#Z countable infinity c = \#R the continuum f = \#\{g| g: [0, 1]->R\} In this domain, these values are obtained using a := Aleph 0, c := 2^a, f := 2^c.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: (%, %) -> Union(%, failed)
x - y returns an element z such that z+y=x or “failed” if no such element exists.
<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
^: (%, %) -> %
x^y returns \#(X^Y) where X^Y is defined as \{g| g: Y->X\}.
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
Aleph: NonNegativeInteger -> %
Aleph(n) provides the named (infinite) cardinal number.
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: NonNegativeInteger -> %
from RetractableTo NonNegativeInteger
countable?: % -> Boolean
countable?(a) determines whether a is a countable cardinal, i.e. an integer or Aleph 0.
finite?: % -> Boolean
finite?(a) determines whether a is a finite cardinal, i.e. an integer.
generalizedContinuumHypothesisAssumed: Boolean -> Boolean
generalizedContinuumHypothesisAssumed(bool) is used to dictate whether the hypothesis is to be assumed.
generalizedContinuumHypothesisAssumed?: () -> Boolean
generalizedContinuumHypothesisAssumed?() tests if the hypothesis is currently assumed.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
max: (%, %) -> %
from OrderedSet
min: (%, %) -> %
from OrderedSet
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
recip: % -> Union(%, failed)
from MagmaWithUnit
retract: % -> NonNegativeInteger
from RetractableTo NonNegativeInteger
retractIfCan: % -> Union(NonNegativeInteger, failed)
from RetractableTo NonNegativeInteger
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from MagmaWithUnit
smaller?: (%, %) -> Boolean
from Comparable
zero?: % -> Boolean
from AbelianMonoid

AbelianMonoid

AbelianSemiGroup

BasicType

CoercibleTo OutputForm

CommutativeStar

Comparable

Magma

MagmaWithUnit

Monoid

OrderedSet

PartialOrder

RetractableTo NonNegativeInteger

SemiGroup

SetCategory