\documentclass[12pt]{article}
\newcommand{\vs}{\vspace{.1in}}

\begin{document}
{\bf Stony Brook University ~~~~ MAT 336 ~~~~ History of Mathematics}
\vs

\begin{center}
{\bf On an elementary question in set theory}

Georg Cantor

{\small [Jaresbericht der Deutsch. Math. Vereing. {\bf 1} 75-78 (1890-91)]}
\end{center} 
\vs

In the article entitled: On a property of the set of all real algebraic
numbers (Journ. Math. {\bf 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 $1, 2, 3, \dots ,
\nu , \dots $ or, as I put it, do not have the cardinality of the number
sequence $1, 2, 3, \dots , \nu,  \dots $. From what is proved in \S 2
there follows in fact something further, that for example the set of
all real numbers in an arbitrary interval $(\alpha\dots\beta)$ {\em may not}
be represented as a sequence
$$\omega_1, \omega_2, \dots, \omega_{\nu}, \dots  .$$

Each of these propositions can be given a much more simple proof, which
is independent of considerations about the irrational numbers.

Specifically, let $m$ and $w$ be two different symbols, and let us consider 
the set $M$ of elements $$E = (x_1, x_2, \dots, x_{\nu}, \dots ),$$
which depend on infinitely many coordinates $x_1, x_2, \dots, x_{\nu}, \dots$,
where each of these coordinates is either $m$ or $w$. Let $M$ be the set
of all such elements $E$.

Among the elemets of $M$ we find for example the following three:

$$E^I = (m, m, m, m, \dots),$$
$$E^{II} = (w, w, w, w, \dots),$$
$$E^{III} = (m, w, m, w, \dots).$$

I now state that such a set $M$ does not have the cardinality
of the sequence $1, 2, 3, \dots , \nu , \dots $.

This is a consequence of the following proposition:

``Let $E_1, E_2, \dots, E_{\nu}, \dots $ be any infinite sequence
of elements of the set $M$; then there is an element $E_0$
of $M$ which does not coincide with any $E_{\nu}$.''

For the proof let 

$$E_1  = (a_{11}, a_{12}, \dots , a_{1\nu}, \dots),$$
$$E_2  = (a_{21}, a_{22}, \dots , a_{2\nu}, \dots),$$
$$\dots$$
$$E_{\mu}  = (a_{\mu 1}, a_{\mu 2}, \dots , a_{\mu\nu}, \dots),$$
$$\dots$$

Here each $a_{\mu\nu}$ is set to be $m$ or $w$. We will now
define a sequence $b_1, b_2, \dots, b_{\nu}, \dots $ in such a way
that $b_{\nu}$ is also only $m$ or $w$ and is {\em different} from
$a_{\nu\nu}$.

So if $a_{\nu\nu} = m$ then $ b_{\nu} = w$, and
if $a_{\nu\nu} = w$ then $ b_{\nu} = m$.

Let us now consider the  element
$$E_0 = (b_1, b_2, b_3, \dots )$$
of $M$; we immediately see that the equation
$$E_0 = E_{\mu}$$
cannot be satisfied for any integer value of $\mu$, because
for that particular $\mu$ and for all integer values of $\nu$
we would have
$$b_{\nu} = a_{\mu,\nu},$$ 
and  in particular
$$b_{\mu} = a_{\mu,\mu},$$ 
which is excluded by the definition of $b_{\mu}$. From this
proposition it follws immediately that the set of all the elements
of $M$ cannot be placed in a sequence: $E_1, E_2, \dots, E_{\nu}, \dots $,
since we would be faced with the contradiction that the object $E_0$
would be both an element of $M$ and not an element of $M$.

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 $L$
we can find another set $M$ of larger cardinality than $L$.

For example, let $L$ be an interval, say the set of all
real numbers which are $\ge 0$ and $\le 1$.

Then let $M$ be the set of all functions $f(x)$ which only
take on the two values $0$ and $1$, while $x$ runs through all
the real values $\ge 0$ and $\le 1$.

That $M$ {\em does not have a smaller} cardinality than $L$ follows
from the fact that $M$ has subsets which are of the same cardinality
as $L$, for example the subset consisting of the functions of $x$
which give $1$ just for a single value $x_0$, and $0$ for all other
values of $x$.

But $M$ {\em cannot have the same} cardinality as $L$, because if it
did then the elements of $M$ could be put in one-one correspondence
with the variable $z$, and $M$ could be thought of as  a function
of the two variables $x$ and $z$

$$\varphi(x,z)$$

\noindent
so that each choice of $z$ would give the element $f(x) = \varphi(x,z)$
of $M$ and vice-versa each element $f(x)$ of $M$ would correspond to
$\varphi(x,z)$ for some choice of $z$. But this leads to a contradiction.
Because then let us consider the function $g(x)$ which only takes on
the values $0$ and $1$, and which for each $x$ is different from
$\varphi(x,x)$; then on the one hand $g(x)$ is an element of $M$,
and on the other hand $g(x)$ cannot be $\varphi(x,z)$ for any
choice $z = z_0$, because $\varphi(z_0,z_0)$  is different from
$g(z_0)$.

Since the cardinality of $M$ is neither smaller than or the same
as that of $L$, it must be larger than the cardinality of $L$.
(see Crelles Journal {\bf 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.
\vspace{1in}

{\em Translator's note:} 

\begin{itemize}
\item This is a rough translation. I have tried to keep the flavor
of the original, but I used ``set'' for Cantor's {\em Mannigfaltigkeit} (``manifold''),
{\em Gesamtheit} (``totality''), {\em Inbegriff} (``clutch'') and {\em Menge}, now standard for ``set,'' which appears twice, in the last two
paragraphs. It is interesting that the
notations $(a,b)$ and $[a,b]$ for open and closed intervals were
not available to Cantor, nor was the possibility of writing $f$
without $(x)$.
\item Zermelo, the editor of the collected works where this article was
located, remarks: That the set of all cardinalities is a well-ordered
system (even if not a ``set'') is in no way ``proven'' in the referenced
work; Cantor does not yet have the proof that every set can be
well-ordered and therefore that every cardinality is an Aleph.
\end{itemize}

Anthony Phillips









\end{document}