Ordinal Numbers Vinay Singh MARCH 20, 2012

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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 NumbersVinay SinghMARCH 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)
• 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

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-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
• 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

Hassediagram of a Power Set

Partial Order

Total Order

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
• 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
• 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?