Lecture 5 Infinite Ordinals
Recall: What is “2”? Definition: 2 = {0,1}, where 1 = {0} and 0 = {}. (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 Since n = {0,1,2,…,n 1}, n+1 = {0,1,2,…,n 1,n} = {0,1,2,…,n 1} {n} = n {n} Thus, we have the following recursive definition of the natural numbers: Base: 0 = {} Step: n+1 = n {n}
The Infinite Ordinal For n,m N, (n m n < m) Thus, we actually defined the order structures: (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 Note that (N,<) = ( , ) is linearly ordered, i.e. ( n,m N)(n < m or n = m or m < n) Moreover, the order (N,<) has the following nice 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.
But why stop at ? Definition: = {0,1,2,3,…} (= N) = 0 1 2 3 … +1 = { } = {0,1,2,3,…, } +2 = ( +1)+1 = ( +1) { +1} = {0,1,2,3,…, , +1} +3 = ( +2)+1 = ( +2) { +2} = {0,1,2,3,…, , +1, +2}... + = ( +1) ( +2) ( +3) … = {0,1,2,3,…, , +1, +2, +3,…} = 2
And continue… Definition: 2 = {0,1,2,3,…, , +1, +2, +3,…} 2+1 = {0,1,2,3,…, , +1, +2, +3,…, 2} 2+2 = {0,1,2,…, , +1, +2,…, 2, 2+1} … 2+ = {0,1,2,…, , +1, +2,…, 2, 2+1,…} = 3 … = 2 = {0,1,…, ,…, 2,…, 3,…}
And continue… Definition: 2 = {0,1,…, ,…, 2,…, 3,…} 2 +1 = {0,1,…, ,…, 2,…, 3,…, 2 } … 2 + = {0,1,…, 2, 2 +1,…} … 2 + 2 = {0,…, 2,…, 2 + ,…, 2 + 2,…} = 2 2 … 2 = {0,…, 2,…, 2 2,…, 2 3,…} = 3
And continue… Definition: 3 = {0,1,…, 2,…, 2 2,…, 2 3,…} 3 +1 = {0,1,…, 2,…, 2 2,…, 2 3,…, 3 } … 3 + = {0,1,…, 3, 3 +1,…} … 3 + 3 = {0,…, 3,…, 3 + 2,…, 3 + 2 2,…} = 3 2 … 4 ; … ; 5 ; … ; ; … ; ; … ; 0 = ... ; …
Ordinals versus Cardinals Notes: Cardinals measure sizes of sets Ordinals measure lengths of well ordered sets All ordinals mentioned so far, e.g. and 0 = ... are actually countable sets. There are however uncountable ordinals, 1 is the least uncountable ordinal. In fact, 1 = the set of all countable ordinals. Need to define what ordinals really are.
More rigorously Definition: An ordinal is a set X such that: X is linearly ordered by , i.e. ( 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 Definition: Ordinals can be classified into three classes: The ordinal 0 Successor ordinals = + 1 = { } Limit ordinals = sup{ : < } = { : < } Definitions of ordinal functions according to this classification are said to use transfinite recursion. Proofs of ordinal statements according to this classification are said to use transfinite induction.
Ordinal Arithmetic: Addition Definition: We define the sum of two ordinals + by recursion on : Base ( = 0): + 0 = Successor ( = + 1): + ( + 1) = ( + ) + 1 = ( + ) {( + )} Limit ( = sup{ : < }): + = sup{ + : < } Note:. The definition generalizes the addition of natural numbers. Example: 1+ = < +1, so ordinal addition is not commutative, i.e. + + , in general.
Ordinal Arithmetic: Multiplication Definition: We define the product of two ordinals 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 Definition: We define the exponentiation of two 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