“ The transfinite numbers are in a sense new-irrationalities ,” 109 and so , according to Cantor , one cannot be consistent in accepting irrational numbers and rejecting the transfinite numbers .
Cantor ’ s Transfinite Numbers . The first transfinite number , ω , is the limiting point of the natural numbers ; it is not a natural number but a transfinite number . It stands beyond all
natural numbers similarly to the number 1 standing beyond all fractions of the form n−1 n . Formally , ω is the collection of all natural numbers , i . e ., the set { 0 , 1 , 2 , 3 , 4 , … }. In formal set theory , natural numbers are defined in terms of the empty set . The empty set , ∅ , gives rise to the number 0 for the empty set has no elements . The set containing the empty set , { ∅ }, has one element ; the set { ∅ , { ∅ }} has two elements , and so on . In other words , 0 = ∅ , 1 = { 0 }, 2 = { 0 , 1 }, 3 = { 0 , 1 , 2 }, etc … and ω is the collection of the natural numbers : ω = { 0 , 1 , 2 , 3 , … }. Positing the existence of a new kind of number , greater than all natural numbers , does not lead to a contradiction . Moreover , extending the notion of number is not completely novel because of
109
Ibid , 77 . Page 43 of 62