Vinay Singh MARCH 20 2012 MAT 7670 Introduction to Ordinal Numbers Ordinal Numbers Is an extension domain of Natural Numbers ℕ different from Integers ℤ and Cardinal numbers Set sizing ID: 181797
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Slide1
Ordinal NumbersVinay SinghMARCH 20, 2012
MAT 7670Slide2
Introduction to Ordinal NumbersOrdinal NumbersIs an extension (domain ≥) of Natural Numbers (ℕ) different from Integers (ℤ) and Cardinal numbers (Set sizing)
Like
other kinds of numbers, ordinals can be added, multiplied, and
even
exponentiated
Strong applications to topology (continuous deformations of shapes)
Any ordinal number can be turned into a
topological space
by using the
order
topology
Defined
as the
order type
of a
well-ordered set
.Slide3
Brief HistoryDiscovered (by accident) in 1883 by Georg Cantor to classify sets with certain order structures
Georg
Cantor
Known as the inventor of Set Theory
E
stablished the
importance of
one-to-one
correspondence between the members of two
sets (
Bijection
)
Defined infinite and well-ordered sets
Proved that real numbers are “more numerous” than the natural numbers
…Slide4
Well-ordered SetsWell-ordering on a set S is a total order on S where every non-empty subset has a least element
Well-ordering theorem
Equivalent
to the axiom of
choice
States
that every set can be well-ordered
Every well-ordered set is order isomorphic (has the same order) to a unique ordinal numberSlide5
Total Order vs. Partial OrderTotal OrderAntisymmetry - a ≤ b and b ≤
a
then
a
=
b
Transitivity -
a ≤ b and b ≤ c then a ≤ cTotality - a ≤ b or b ≤
a
Partial Order
Antisymmetry
Transitivity
Reflexivity -
a
≤ aSlide6
Ordering Examples
Hasse
diagram of a Power Set
Partial Order
Total OrderSlide7
Cardinals and Finite OrdinalsCardinalsAnother extension of ℕOne-to-One correspondence with ordinal numbers
Both finite and infinite
Determine size of a set
Cardinals – How many?
Ordinals – In what order/position?
Finite Ordinals
Finite ordinals are (equivalent to) the natural numbers (0, 1, 2, …)Slide8
Infinite OrdinalsInfinite OrdinalsLeast infinite ordinal is ωIdentified by the cardinal number ℵ0(Aleph Null)
(Countable vs. Uncountable)
Uncountable many countably infinite ordinals
ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω
2
, …, ω
3
, …, ωω, …, ωωω, …, ε0, ….Slide9
Ordinal ExamplesSlide10
Ordinal ArithmeticAdditionAdd two ordinals Concatenate
their order
types
Disjoint sets S and T can be added by taking the
order type of
S∪T
Not commutative ((
1+ω = ω) ≠ ω+1)MultiplicationMultiply two ordinalsFind the Cartesian Product S
×
T
S
×
T can be
well-ordered by taking the variant lexicographical order
Also not commutative ((
2
*
ω
=
ω
)
≠
ω
*
2
)
Exponentiation
For finite exponents, power is iterated multiplication
For infinite exponents, try not to think about it unless you’re Will Hunting
For
ω
ω
, we can try to visualize the set of infinite sequences of
ℕSlide11
Questions
Questions?