Download - Ordinal Numbers Vinay Singh MARCH 20, 2012
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?