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## PowerPoint Slideshow about ' Ordinal Numbers Vinay Singh MARCH 20, 2012' - loki

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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.

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

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

- 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?

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