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
sequence
. 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
may not
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.
Translator's note:
Anthony Phillips