the previous extensions of the concept of number to irrational and complex numbers . The fundamental question of theory of the transfinite numbers is whether ω is the only infinite number . Contrary to ancient and medieval intuition , which posed an undifferentiated notion of the infinite , ω is not the only transfinite number . In fact , Cantor proved that there are infinitely many transfinite numbers . 110 His proof showed that infinite numbers could be differentiated and studied mathematically .
Cantor distinguishes two types of transfinite numbers . The ordinal numbers are constructed by a limiting process from natural numbers with ω being the first transfinite number which is followed by ω + 1 , ω + 2 , ω + 3 , … The limiting process applied to ordinal numbers ω + n yields ω + ω which can be also written as 2ω . The process of construction of ordinal numbers can be continued indefinitely , producing 2ω , 3ω ,… etc ., and then ω ∙ ω = ω 2 , ω 3 ,… etc ... The elements of ordinal numbers are governed by a linear order , and what distinguishes ω and ω + 1 is not their size , for they , in fact , have the same size , 111 but the existence of the largest element in ω + 1 which is lacking in ω . The second type of transfinite numbers is the cardinal numbers . Cantor chose to denote the infinite cardinal numbers with the Hebrew letter aleph : ℵ 0 , ℵ 1 , ℵ 2 , … with ℵ 0 = ω . Cardinal numbers are ordinal numbers that cannot be placed in oneto-one correspondence with any smaller ordinal numbers ; hence , ℵ 1 is the smallest ordinal
110
Dauben , " Georg Cantor ,” 4 .
111
Two infinite sets are shown to have the same size not by counting their elements which is impossible but by showing that there is a one-to-one correspondence between their elements . Two sets A and B are of the same size if there exists a function f : A→B such that every element of set A is assigned to one element from set B , all elements of A and B are used , and every element of B is used exactly once . It is easy to see that for finite sets the existence of such a function is equivalent to the two sets having the same number of elements . The advantage of the one-to-one correspondence over counting the elements is its applicability to sets than cannot be counted ; and hence to infinite sets . Two sets are said to have the same number of elements if there exists a one-to-one correspondence between them . The one-to-one correspondence function does not establish the size of the sets but its existence determines that the sets have the same size ; at the same time , it is logically more fundamental and prior to counting . See ( Russell , Definition of Number 1963 ).
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