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