Cantor ’ s response to the impossibility of adding to an infinite . Cantor gave a complete description of addition , multiplication and exponentiation of ordinal types , ordinal numbers and cardinal numbers . Suppose that the series of the past days has the ordinal type of ω ∗ , as Aquinas suggests . Since ̅̅̅̅̅̅̅̅̅ ω ∗ + 1 = ω ∗ , adding the present day to the series of the past days does not change the ordinal type of the series of the past days . This will be true even if the series of the past days is found to have a more complex ordinal type as long as it ends with ω ∗ , which is necessary . Hence , Cantor agrees with Aquinas that “ an addition can be made to the infinite but not in a way that would make one infinite greater than another infinite .” 119
On the other hand , Cantor disagrees with Aquinas on the impossibility of adding a day on the infinite side of the series of the past days . If there were such an additional day , the ordinal type would become 1 + ω ∗ which is different from ω ∗ . In that case , the infinite would change its ordinal type . Moreover , Cantor would not hesitate to add an additional day to any of the ordinal types . Let ’ s take into consideration the remaining possible ordinal types of the series of the past
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Robin Small , " Cantor and the Scholastics ," American Catholic Philosophical Quarterly 66 , no . 4 ( 1992 ): 415 . Page 49 of 62