nLab Beiträge zur Begründung der transfiniten Mengenlehre

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• Georg Cantor,

  Beiträge zur Begründung der transfiniten Mengenlehre,

  Math. Ann. 46 (1895) pp.481-512,

  reprinted from p. 282 on in

  Ernst Zermelo (ed.),

  Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts,

  Springer Berlin 1932

  (online English translation)

which is the origin of the modern concept of cardinality of sets. (But see also the commentary in William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen").

Section 1. The conception of Power or Cardinal Number**

By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) $M$ of definite and separate objects of our intuition or our thought. These objects are called the “elements” of $M$. In signs we express this thus :

(i) $M = \{m\}$.

We denote the uniting of many aggregates $M$, $N$, $P$, $\cdots$, which have no common elements, into a single aggregate by

(2) $(M, N, P, \cdots)$.

The elements of this aggregate are, therefore, the elements of $M$, of $N$, of $P$, …, taken together. We will call by the name “part” or “partial aggregate ” of an aggregate M any other aggregate $M_1$ whose elements are also elements of $M$. If $M_2$ is a part of $M_1$ and $M_1$ is a part of $M$, then $M_2$ is a part of $M$.

Every aggregate $M$ has a definite “power”, which we will also call its “cardinal number”.

We will call by the name “power” or “cardinal number” of $M$ the general concept which, by means of our active faculty of thought, arises from the aggregate $M$ when we make abstraction of the nature of its various elements $m$ and of the order in which they are given.

[482] We denote the result of this double act of abstraction, the cardinal -number or power of M, by (3) $\overline{\overline{M}}$

Since every single element $m$, if we abstract from its nature, becomes a “unit,” the cardinal number $M$ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate $M$. We say that two aggregates $M$ and $N$ are “equivalent,” in signs

(4) $M \sim N$ or $N \sim M$

if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. To every part $M_1$ of $M$ there corresponds, then, a definite equivalent part $N_1$ of $N$, and inversely.

$[$ …$]$

Every aggregate is equivalent to itself

(5) $M \sim M$

If two aggregates are equivalent to a third, they are equivalent to one another ; that is to say :

(6) from $M \sim P$ and $N \sim P$ follows $M \sim N$.

Of fundamental importance is the theorem that two aggregates $M$ and $N$ have the same cardinal number if, and only if, they are equivalent : thus,

(7) from $M \sim N$ we get $\overline{\overline{M}} = \overline{\overline{N}}$,

and

(8) from $\overline{\overline{M}} = \overline{\overline{N}}$ we get $M \sim N$.

Thus the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.

[483] In fact, according to the above definition of power, the cardinal number $M$ remains unaltered if in the place of each of one or many or even all elements $m$ of $M$ other things are substituted. If, now, $M \sim N$, there is a law of co-ordination by means of which $M$ and $N$ are uniquely and reciprocally referred to one another; and by it to the element $m$ of $M$ corresponds the element $n$ of $N$. Then we can imagine, in the place of every element $m$ of $M$, the corresponding element $n$ of $N$ substituted, and, in this way, $M$ transforms into $N$ without alteration of cardinal number. Consequently

$\overline{\overline{M}} = \overline{\overline{N}}$.

The converse of the theorem results from the remark that between the elements of $M$ and the different units of its cardinal number $M$ a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, $\overline{\overline{M}}$ grows, so to speak, out of $M$ in such a way that from every element $w$ of $M$ a special unit of $M$ arises. Thus we can say that

(9) $M \sim \overline{\overline{M}}$

In the same way $N \sim \overline{\overline{N}}$. If then $\overline{\overline{M}} = \overline{\overline{N}}$, we have, by (6), $M \sim N$.

We will mention the following theorem, which results immediately from the conception of equival ence. If $M$, $N$, $P$, … are aggregates which have no common elements, $M'$, $N'$, $P'$, … are also aggregates with the same property, and if

$M \sim M'$, $N \sim N'$, $P \sim P'$, …,

then we always have

$(M, N, P, . . .) \sim (M', N', P', \cdots)$.

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