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Ordinal Numbers Vinay Singh MARCH 20, 2012

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

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  1. Ordinal NumbersVinay SinghMARCH 20, 2012 MAT 7670

  2. 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.

  3. 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 • …

  4. 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

  5. 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

  6. Ordering Examples Hassediagram of a Power Set Partial Order Total Order

  7. 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, …)

  8. 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, ….

  9. Ordinal Examples

  10. 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

  11. Questions Questions?

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