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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,…,n1} = {kN: 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,…,n1}, n+1 = {0,1,2,…,n1,n} = {0,1,2,…,n1}{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,mN, (n m n < m) Thus, we actually defined the order structures: (n,<) = (n,) On each n, is a transitive relation, i.e. (i,j,kN)(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,mN)(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 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 ? Definition: = {0,1,2,3,…} (= N) = 0123… +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,…} = 22 … 2 = {0,…,2,…,22,…,23,…} = 3 And continue… Definition: 3 = {0,1,…,2,…,22,…,23,…} 3+1 = {0,1,…,2,…,22,…,23,…,3} … 3+ = {0,1,…,3,3+1,…} … 3+3 = {0,…,3,…,3+2,…,3+22,…} = 32 … .. . 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,wX)(y z and z w y w) and (y,zX)(y z or y = z or z y) X is transitive, i.e. (yX)(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 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