days which we mentioned in the previous section : 1 + ω ∗ , ω + ω ∗ , and ω ∗ + ω + ω ∗ . Adding one day on the infinite side of each of these ordinal types affects all of them except for ω + ω ∗ because 1 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + ω + ω ∗ = ω + ω ∗ .
Bonaventure ’ s comparison of the past solar revolutions to the past lunar revolutions , can be clarified by multiplication of ordinal types . The solar revolutions have the ordinal type of ω ∗ = {… ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ , s 3 , s 2 , s 1 } . Since for every solar revolution , there are twelve lunar revolutions , lunar revolutions also have the ordinal type of ω ∗ ∙ 12 . Each solar revolution , s n , corresponds to twelve lunar revolutions l 1 , l 2 , − , l 12 , so the series of the solar revolutions {… , s 3 , s 2 , s 1 } leads to the series of lunar revolutions that can be represented by : {… , l 3 , 1 , l 3 , 2 , − , l 3 , 12 , l 2 , 1 , l 2 , 2 , − , l 2 , 12 , l 1 , 1 , l 1 , 2 , − , l 1 , 12 }. It is clear that this series is also of ω ∗ ordinal type . However , multiplication of ordinal types in not commutative . For an example , the ordinal type of 12 ∙ ω ∗ is not the same as ω ∗ , but consists in twelve copies of ω ∗ and cannot be simplified .
The above considerations show that Cantor ’ s study of infinite sets leads to a deeper understanding of the distinction between numbers and pluralities . This distinction is already present in Aristotle : “ number is not just a plurality ; it is plurality measured by some unit .” 120 Numbers designate only one aspect of the set , namely its size but say nothing about the reality of its elements or their structure . In assigning a number to a set , we are concerned only with one abstract aspect of the set . “ For this reason , assigning the same number to two sets does not mean attributing ‘ one and the same reality ’ to them .” 121 For an example , the set of ten men is different
120
Hippocrates George Apostle , Aristotle ' s Philosophy of Mathematics ( Chicago : The University of Chicago Press , 1952 ), 90 .
121
Small , " Cantor and the Scholastics ," 416 . Page 50 of 62