Georg Cantor

“ We can, for instance, always take care that to a special element of a special element of corresponds. For if, according to the original law, the elements and do not correspond to one another, but to the element of the element of corresponds, and to the element of the element of corresponds, we take the modified law according to which corresponds to and to and for the other elements the original law remains unaltered. By this means the end is attained.
Every aggregate is equivalent to itself:
(5)
If two aggregates are equivalent to a third, they are equivalent to one another ”

“ That class which naturally presents itself first to us, and whose complete investigation must, accordingly, be the next special aim of the theory of transfinite aggregates, is the class of types which embraces all the types with the least transfinite cardinal number . From the cardinal number which determines the class of types we have to distinguish that cardinal number which for its part [499] is determined by the class of types . ”

“ It contains nothing arbitrary, but is the natural extension of the concept of number. It deserves to be especially emphasized that the criterion of equality (4) follows with absolute necessity from the concept of ordinal type and consequently permits of no alteration. The chief cause of the grave errors in G. Veronese's Grundzüge der Geometrie (German by A. Schepp, Leipzig, 1894) is the non-recognition of this point. ”

“ It may happen that , as, for example, in the case of finite types or in that of the type of the aggregate of all rational numbers which are greater than and less than in their natural order of precedence. ”

“ By a "covering of the aggregate with elements of the aggregate ," or, more simply, by a "covering of with ," we understand a law by which with every element of a definite element of is bound up, where one and the same element of can come repeatedly into application. The element of bound up with is, in a way, a one-valued function of , and may be denoted by ”

“ We see how pregnant and far-reaching these simple formulæ extended to powers are by the following example. If we denote the power of the linear continuum (that is, the totality of real numbers such that and ) by , we easily see that it may be represented by, amongst others, the formula:
(11)
,
where § 6 gives the meaning of . In fact, by (4) , is the power of all representations ”

“ Such a correspondence of similar aggregates we call an "imaging" (Abbildung) of these aggregates on one another. In such an imaging, to every part—which obviously also appears as an ordered aggregate— of corresponds a similar part of .
We express the similarity of two ordered aggregates and by the formula:
(3)
. ”

“ B. Every one of these numbers is greater than the preceding ones and less than the following ones (§ 2) .
C. There are no cardinal numbers which, in magnitude, lie between two consecutive numbers and (§ 2) . ”