In the article entitled: On a property of the set of all real algebraic
numbers (Journ. Math. 77 258) there was presented for the first time
a proof that there are infinite sets which cannot be put into one-one
correspondence with the set of all finite whole numbers
or, as I put it, do not have the cardinality of the number
. From what is proved in §2
there follows in fact something further, that for example the set of
all real numbers in an arbitrary interval
be represented as a sequence
Each of these propositions can be given a much more simple proof, which is independent of considerations about the irrational numbers.
Specifically, let and be two different symbols, and let us consider
the set of elements
Among the elemets of we find for example the following three:
I now state that such a set does not have the cardinality of the sequence .
This is a consequence of the following proposition:
``Let be any infinite sequence of elements of the set ; then there is an element of which does not coincide with any .''
For the proof let
Here each is set to be or . We will now define a sequence in such a way that is also only or and is different from .
So if then , and if then .
Let us now consider the element
This proof is remarkable not only because of its great simplicity, but also because the principle it contains leads immediately to the general proposition that the cardinalities of well defined sets have no maximum; or, equivalently, that for any given set we can find another set of larger cardinality than .
For example, let be an interval, say the set of all real numbers which are and .
Then let be the set of all functions which only take on the two values and , while runs through all the real values and .
That does not have a smaller cardinality than follows from the fact that has subsets which are of the same cardinality as , for example the subset consisting of the functions of which give just for a single value , and for all other values of .
But cannot have the same cardinality as , because if it did then the elements of could be put in one-one correspondence with the variable , and could be thought of as a function of the two variables and
so that each choice of would give the element of and vice-versa each element of would correspond to for some choice of . But this leads to a contradiction. Because then let us consider the function which only takes on the values and , and which for each is different from ; then on the one hand is an element of , and on the other hand cannot be for any choice , because is different from .
Since the cardinality of is neither smaller than or the same as that of , it must be larger than the cardinality of . (see Crelles Journal 84 242).
I have already, in ``Foundations of a general theory of sets'' (Leipzig 1883; Math. Ann. Vol. 21) shown, by completely different techniques, that the cardinalities have no maximum; there it is also proved that the set of all cardinalities, when we think of them as ordered by their size, forms a ``well-ordered set'' so that in nature for every cardinality there is a next larger, but also every infinite set of cardinalities is followed by a next larger.
The ``cardinalities'' represent the single and remarkable generalization of the finite ``cardinal numbers;'' they are nothing else but the actual-infinitely-large cardinal numbers, and they inherit the same reality and definiteness as those do; only the relations between them form a different ``number theory'' from the finite one.
The further completion of this field is a job for the future.