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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 numbers vinay singh march 20 2012

Ordinal NumbersVinay SinghMARCH 20, 2012

MAT 7670


Introduction to ordinal numbers
Introduction to Ordinal Numbers

  • 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.


Brief history
Brief History

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 ordered sets
Well-ordered Sets

  • 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 vs partial order
Total Order vs. Partial Order

  • 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


Ordering examples
Ordering Examples

Hassediagram of a Power Set

Partial Order

Total Order


Cardinals and finite ordinals
Cardinals and Finite Ordinals

  • 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
Infinite Ordinals

  • 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, ….



Ordinal arithmetic
Ordinal Arithmetic

  • Addition

    • Add 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)

  • 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
Questions

Questions?