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 HistoryDiscovered (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

Hasse diagram 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 Examples

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?