natural numbers but not of the negative whole numbers nor the rational numbers . The chapters 7 through 11 of Cantor ’ s Contributions to the Founding of the Theory of Transfinite Numbers are dedicated to the study of various ordinal types with the emphasis given to the ordinal type of rational numbers .
Ordinal numbers are a particular kind of ordinal types that arise when the original set is well-ordered . A set is well-ordered if each of its subsets has the smallest element . For an example , the set of natural numbers is well-ordered but the set of rational numbers is not . Cardinal numbers , in turn , are identified with the ordinal numbers that cannot be placed in a oneto-one correspondence with any smaller ordinal number . For an example , ω is a cardinal number because it cannot be put in one-to-one correspondence with any finite number . However , ω + ω is not a cardinal number for its elements : { a 1 , a 2 , a 3 , … , b 1 , b 2 , b 3 , … } can be rearranged and put in one-to-one correspondence with ω : { a 1 , b 1 , a 2 , b 2 , … }. Therefore , cardinal numbers are intrinsically well-ordered . The well-ordering of the cardinal numbers comes as a surprise when one keeps in mind Cantor ’ s initial definition of cardinal numbers : “ the general concept which , by means of our active faculty of thought , arises from the [ set ] when we make abstraction of the nature of its various elements and of the order in which they were given .” 116 Cantor denotes the cardinal number of a given set M as M̿ in order to signify the double act of abstraction which leaves behind not only the nature of the particular elements of M but also the order of precedence among them . While the double abstraction leaves behind the original ordering of the set , it imposes a new well-ordered structure on its abstracted elements .
The notion of order is fundamental to Cantor ’ s mathematical thinking . In fact , in his first major publication of the theory of the transfinite numbers in 1883 , Cantor regarded the well-
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Ibid , 86 . Page 47 of 62