Page:Cantortransfinite.djvu/136

Rh the first the element $$f_{\nu_0'+\nu'}$$ of the second corresponds. Since $$\nu_0'$$ is arbitrary, we have here an infinity of imagings.

The concept of "ordinal type" developed here, when it is transferred in like manner to "multiply ordered aggregates," embraces, in conjunction with the concept of "cardinal number" or "power" introduced in §1, everything capable of being numbered (Anzahlmässige) that is thinkable, and in this sense cannot be further generalized. 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.

On page 30 the "number (Anzahl oder Zahl) of an ordered group" is defined in exactly the same way as what we have called the "ordinal type of a simply ordered aggregate" (Zur Lehre vom Transfiniten, Halle, 1890, pp. 68-75; reprinted from the Zeitschr. für Philos. und philos. Kritik for 1887). [501] But Veronese thinks that he must make an addition to the criterion of equality. He says on page 31: "Numbers whose units correspond to one another uniquely and in the same order and of which the one is neither a part of the other nor equal to a part of the other are