/
Ordinal Numbers Ordinal Numbers

Ordinal Numbers - PowerPoint Presentation

cheryl-pisano
cheryl-pisano . @cheryl-pisano
Follow
444 views
Uploaded On 2015-11-03

Ordinal Numbers - PPT Presentation

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

ordinals order numbers ordinal order ordinals ordinal numbers set infinite ordered sets finite total ordering partial natural cardinals number

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "Ordinal Numbers" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

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?