ordering principle that states : “ it is always possible to bring every well-defined set into the form of a well-ordered set ” 117 to be a self-evident fundamental law of thought leading to rich conclusions . It is not surprising that Bertrand Russell summarized the modern developments in mathematics in the following words : “ Mathematics has , in modern times , brought order into greater and greater prominence . In former days , it was supposed ( and philosophers are still apt to suppose ) that quantity was the fundamental notion of mathematics . But nowadays , quantity is banished altogether (…) while order more and more reigns supreme .” 118 Cantor considered order to be as fundamental to mathematics as , if not more than , the notion of the unit .
What would Cantor reply to the proposed impossibility of ordering an infinite ? Firstly , every set can be not only ordered but well-ordered regardless of its initial structure . However , in the consideration of an infinite series of the past days , we deal with a series naturally ordered by a simple order . Hence , secondly , Cantor would try to identify the possible ordinal type of the series of the past days . According to Aquinas , its ordinal type should be ω ∗ , which is the type of negative whole numbers , i . e ., the order in which there is no first element but there is the last element and each element has a unique element prior to it . According to Bonaventure , the ordinal type is either 1 + ω ∗ or ω + ω ∗ . Is it possible that the ordinal type of the past days is different ? Could it be ω ∗ + ω + ω ∗ ? Cantor ’ s difficulty in treating the series of the past days would arise not from an absence of order but from too many possible ordinal types that could be logically assigned to the series of the past days .
117
Akihiro Konamori , " The Mathematical Development of Set Theory from Cantor to Cohen ," The Bulletin of Symbolic Logic 2 , no . 1 ( 1996 ): 5 .
118
Bertrand Russell , “ Mathematics and the Metaphysician ,” in Gateway to the Great Books , vol . 9 . Mathematics , ed . Robert M . Hutchins , and Mortimer J . Adler , 95-110 ( Chicago : Encyclopedia Britannica Inc ., 1963 ), 106 .
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