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) MM of definite and separate objects of our intuition or our thought. These objects are called the “elements” of MM. In signs we express this thus :

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

We denote the uniting of many aggregates MM, NN, PP, \cdots, which have no common elements, into a single aggregate by

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

The elements of this aggregate are, therefore, the elements of MM, of NN, of PP, …, taken together. We will call by the name “part” or “partial aggregate ” of an aggregate M any other aggregate M 1M_1 whose elements are also elements of MM. If M 2M_2 is a part of M 1M_1 and M 1M_1 is a part of MM, then M 2M_2 is a part of MM.

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

We will call by the name “power” or “cardinal number” of MM the general concept which, by means of our active faculty of thought, arises from the aggregate MM when we make abstraction of the nature of its various elements mm 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) M¯¯\overline{\overline{M}}

Since every single element mm, if we abstract from its nature, becomes a “unit,” the cardinal number MM 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 MM. We say that two aggregates MM and NN are “equivalent,” in signs

(4) MNM \sim N or NMN \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 1M_1 of MM there corresponds, then, a definite equivalent part N 1N_1 of NN, and inversely.


Every aggregate is equivalent to itself

(5) MMM \sim M

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

(6) from MPM \sim P and NPN \sim P follows MNM \sim N.

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

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


(8) from M¯¯=N¯¯\overline{\overline{M}} = \overline{\overline{N}} we get MNM \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 MM remains unaltered if in the place of each of one or many or even all elements mm of MM other things are substituted. If, now, MNM \sim N, there is a law of co-ordination by means of which MM and NN are uniquely and reciprocally referred to one another; and by it to the element mm of MM corresponds the element nn of NN. Then we can imagine, in the place of every element mm of MM, the corresponding element nn of NN substituted, and, in this way, MM transforms into NN without alteration of cardinal number. Consequently

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

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

(9) MM¯¯M \sim \overline{\overline{M}}

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

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

MMM \sim M', NNN \sim N', PPP \sim P', …,

then we always have

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

category: reference

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