Lecture 5
Infinite Ordinals
Recall: What is “2”?
 Definition: 2 = {0,1}, where 1 = {0} and 0 = {}.
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(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:
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(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
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feature:
Every nonempty subset of N has a least element
Equivalently:
There is no infinite sequence x0,x1,x2, x3,…N,
such that … < x3 < x2 < x1 < x0.
Any linear order < with this feature is called a well
order.
But why stop at ?
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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…
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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
 …
..
.
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 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.
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(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 x1,x2,x3,…, such that
…  x3  x2  x1
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
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 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
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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