Page:Cantortransfinite.djvu/131

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112
THE FOUNDING OF THE THEORY

By this we understand the general concept which results from if we only abstract from the nature of the elements , and retain the order of precedence among them. Thus the ordinal type is itself an ordered aggregate whose elements are units which have the same order of precedence amongst one another as the corresponding elements of , from which they are derived by abstraction.

We call two ordered aggregates and "similar" (ähnlich) if they can be put into a biunivocal correspondence with one another in such a manner that, if and are any two elements of and and the corresponding elements of , then the relation of rank of to in is the same as that of to in . 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)
.

Every ordered aggregate is similar to itself.

If two ordered aggregates are similar to a third, they are similar to one another.

[498] A simple consideration shows that two ordered aggregates have the same ordinal type if, and only if, they are similar, so that, of the two formulæ