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)  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 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 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, ….

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

Questions Questions?