Cantor ’ s response to the impossibility of ordering an infinite . The notions of ordinal and cardinal numbers are inspired by Cantor ’ s work with sets of natural , rational and real numbers , all of which possess their own particular order of elements . The mathematical notion of an ordered set is captured in set theory by ordinal types . Firstly , a set M is simply ordered if it possesses a relation of precedence , ≺ , between its elements such that whenever two different elements m 1 and m 2 of M are given , either m 1 ≺ m 2 or m 2 ≺ m 1 , and when m 1 , m 2 , and m 3 are elements of M such that m 1 ≺ m 2 and m 2 ≺ m 3 , then m 1 ≺ m 3 . Sets of natural , rational and real numbers are simply ordered by the relation “ less than .” Secondly , the ordinal type of a simply ordered set M , denoted by M̅ , is “ the general concept which results from M if we only abstract from the nature of elements m , and retain the order of precedence among them .” 115 For an example , the set of natural numbers has the same ordinal type as the set of even natural numbers but differs from that of negative whole numbers or rational numbers . The set of natural numbers possesses the smallest element but not the largest . The same is true of the set of even
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Cantor , Contributions , 111 . Page 46 of 62