Ordering and Well-ordered Sets

Another great gain that can be attributed to the new numbers consists for me in a new concept, unmentioned previously by anyone else, regarding the “ordinal-number” of elements of a well-ordered infinite set.

Since this concept is always expressed by a very specific number in our expanded number realm, provided that the order of the elements of the set (which will shortly be more fully defined) is determined, and since, on the other hand, the concept of ordinal-number receives a direct objective representation within our inner intuition, then by this connection between the concept of cardinal-numbers and ordinal-numbers, the reality of the latter, which I emphasize, is also demonstrated in cases where they are actual-infinite.

A well-ordered set is any well-defined set in which the elements are connected to one another by a certain determinate ordering, according to which there is a first element of the set, and where each individual element (if it is not the last in the ordering) is followed by another, and similarly for any finite or infinite set of elements, there is a definite element which is the immediate successor in the ordering (unless there is nothing following the totality of ordering).

Two “well-ordered” sets are said to be of the same ordinal-number (with reference to their given orderings) if a mutually unambiguous correspondence of them is possible in such a way that if E and F are any two elements of one, and E1 and F1 the corresponding elements of the other, the position of E and F in the ordering of the first set is always in agreement with the position of E1 and F1 in the ordering of the second set, so that if E precedes F in the ordering of the first set, then E1 also precedes F1 in the ordering of the second set.

This correspondence, if it is actually possible, as can easily be seen, is always completely determinate. And since there is always one and only one number α in the extended sequence of numbers, where the numbers preceding it (from 1 onwards) have the same ordinal-number in the natural ordering, it is necessary that the ordinal-number of both of those “well-ordered” sets is exactly equal to α, if α is an infinitely large number, and equal to the number α -1 which precedes the number α, if α is a finite integer.

The essential difference between finite and infinite sets is shown by the fact that a finite set, for every ordering that can be applied given to its elements, presents the same ordinal-number of elements. On the other hand, a set consisting of an infinite number of elements will in general be assigned different ordinal-numbers, according to the ordering that is applied to the elements. As we have seen, the cardinal-number of a set is a property of that set, independent of its ordering, but as soon as one has to deal with infinite sets, the ordinal-number of a set is evidently a factor that in general is dependent on a given ordering of its elements. Nevertheless, even in the case of infinite sets, there is a certain correlation between the cardinal-number of the set and the ordinal-number of its elements that is determined for a given ordering.

If we first take a set that has the cardinal-number of the first class and give the elements any specific ordering so that it becomes a “well-ordered set”, then its ordinal-number is always a specific number of the second number-class and can never be replaced by any other number of any other number-class.

On the other hand, any set of the first cardinal-number can be given an ordering so that its ordinal-number, with reference to this ordering, becomes equal to any selected number of the second number-class. We can also express this in the following way: every set of the cardinal-number of the first number-class can be counted by numbers of the second number-class and only by such numbers, and indeed the set can always be given such an ordering of its elements that it can be counted in this ordering by any given number of the second number-class, and that number indicates the ordinal-number of elements of the set with reference to that ordering.

Analogous rules apply for the sets of higher infinite cardinal-numbers. Hence every well-defined set with the cardinal-number of the second number-class is countable by numbers of the third number-class and only by such numbers, and indeed the set can always be given such an ordering of its elements that it can be countable[Footnote5] in this ordering by an arbitrarily specified number of the third number-class, and this number determines the ordinal-number of elements of the set with reference to that ordering.