1. Lecture 5 Infinite Ordinals

2. 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!

3. 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}

4. 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

5. 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.

6. 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

7. 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,…}

8. 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

9. 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 =   ... ; …

10. 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.

11. 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 

12. 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.

13. 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.

14. 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.

15. 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.

16. Thank you for listening. Wafik