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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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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?

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