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# Ordinal Numbers Vinay Singh MARCH 20, 2012 - PowerPoint PPT Presentation

Ordinal Numbers 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)

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### Ordinal NumbersVinay SinghMARCH 20, 2012

MAT 7670

• Ordinal Numbers

• Is 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.

Discovered (by accident) in 1883 by Georg Cantor to classify sets with certain order structures

• Georg Cantor

• Known as the inventor of Set Theory

• Established 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

• Well-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 number

• Total Order

• Antisymmetry - a ≤ b and b ≤ a then a = b

• Transitivity - a ≤ b and b ≤ c then a ≤ c

• Totality - a ≤ b or b ≤ a

• Partial Order

• Antisymmetry

• Transitivity

• Reflexivity - a ≤ a

Hassediagram of a Power Set

Partial Order

Total Order

• Cardinals

• Another extension of ℕ

• One-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, …)

• Infinite Ordinals

• Least infinite ordinal is ω

• Identified 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, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, ….

• 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)

• Multiplication

• Multiply two ordinals

• Find 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 ωω, we can try to visualize the set of infinite sequences of ℕ

Questions?