Lecture 5 Infinite Ordinals Recall What is 2

Lecture 5 Infinite Ordinals

Recall: What is “ 2”? n Definition: 2 = {0, 1}, where 1 = {0} and 0 = {}. n n n (So 2 is a particular set of size 2. ) In general, we can define: n = {0, 1, 2, …, n 1} = {k N: k < n} This is the so-called “von Neumann” notation. We actually achieved to define the natural numbers as sets. In fact, in mathematics, everything is a set!

A Recursive Definition n Since n = {0, 1, 2, …, n 1}, n n+1 = {0, 1, 2, …, n 1, n} = {0, 1, 2, …, n 1} {n} = n {n} n Thus, we have the following recursive definition of the natural numbers: n Base: 0 = {} n Step: n+1 = n {n}

The Infinite Ordinal n For n, m N, (n m n < m) n Thus, we actually defined the order structures: n n n (n, <) = (n, ) On each n, is a transitive relation, i. e. ( i, j, k N)(i j k i k) Also, 0 1 2 3 … N Definition: = {0, 1, 2, 3, …} = N

Well Ordering n Note that (N, <) = ( , ) is linearly ordered, n i. e. ( n, m N)(n < m or n = m or m < n) n Moreover, the order (N, <) has the following nice n n feature: Every nonempty subset of N has a least element Equivalently: There is no infinite sequence x 0, x 1, x 2, x 3, … N, such that … < x 3 < x 2 < x 1 < x 0. Any linear order < with this feature is called a well order.

Ordinals versus Cardinals n Notes: n Cardinals measure sizes of sets n Ordinals measure lengths of well ordered sets n All ordinals mentioned so far, e. g. . and 0 = are actually countable sets. n There are however uncountable ordinals, 1 is the least uncountable ordinal. n In fact, 1 = the set of all countable ordinals. n Need to define what ordinals really are.

More rigorously n Definition: An ordinal is a set X such that: n X is linearly ordered by , i. e. n n ( y, z, w X)(y z and z w y w) and ( y, z X)(y z or y = z or z y) X is transitive, i. e. ( y X)(y X) Notes: From the axiom of foundation, there is no infinite sequence of sets x 1, x 2, x 3, …, such that … x 3 x 2 x 1 Thus, an ordinal is well ordered by

The Class of Ordinals n Definition: Ordinals can be classified into three classes: n The ordinal 0 n Successor ordinals = + 1 = { } n Limit ordinals = sup{ : < } = { : < } n Definitions of ordinal functions according to this classification are said to use transfinite recursion. n Proofs of ordinal statements according to this classification are said to use transfinite induction.

Ordinal Arithmetic: Addition n Definition: We define the sum of two ordinals + by recursion on : n Base ( = 0): +0= n Successor ( = + 1): + ( + 1) = ( + ) + 1 = ( + ) {( + )} n Limit ( = sup{ : < }): + = sup{ + : < } n Note: . The definition generalizes the addition of natural numbers. n Example: 1+ = < +1, so ordinal addition is not commutative, i. e. + + , in general.

Ordinal Arithmetic: Multiplication n Definition: We define the product of two ordinals n n n by recursion on : Base ( = 0): 0 = 0 Successor ( = + 1): ( + 1) = ( ) + Limit ( = sup{ : < }): + = sup{ : < } Note: . The definition also generalizes the multiplication of natural numbers. Example: 2 = < 2, so ordinal multiplication is not commutative, i. e. , in general.

Ordinal Arithmetic: Exponentiation n Definition: We define the exponentiation of two n n n ordinals by recursion on : Base ( = 0): 0 = 1 Successor ( = + 1): + 1 = Limit ( = sup{ : < }): = sup{ : < } Note: . The definition also generalizes the exponentiation of natural numbers. Example: 2 = , so ordinal exponentiation is not the same as cardinal exponentiation.

Thank you for listening. Wafik