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ORDINAL NUMBERS VINAY SINGH MARCH 20, 2012 MAT 7670

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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.

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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 ……

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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

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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

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Ordering Examples Hasse diagram of a Power Set Partial Order Total Order

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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, …)

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Infinite Ordinals Infinite Ordinals Least infinite ordinal is ω Identified by the cardinal number ℵ 0 (Aleph Null) (Countable vs. Uncountable) Uncountable many countably infinite ordinals ω, ω + 1, ω + 2, …, ω ·2, ω ·2 + 1, …, ω 2, …, ω 3, …, ω ω, …, ω ωω, …, ε 0, ….

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Ordinal Examples

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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 ω ω, we can try to visualize the set of infinite sequences of ℕ

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Questions Questions?

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