# CardinalNumber¶

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
coerce: 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

generalizedContinuumHypothesisAssumed?() tests if the hypothesis is currently assumed.

generalizedContinuumHypothesisAssumed: Boolean -> Boolean

generalizedContinuumHypothesisAssumed(bool) is used to dictate whether the hypothesis is to be 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
retractIfCan: % -> Union(NonNegativeInteger, failed)
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

CommutativeStar

Comparable

Magma

MagmaWithUnit

Monoid

OrderedSet

PartialOrder

SemiGroup

SetCategory

TwoSidedRecip