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one is always a consequence of the other.

If, with an ordinal type ${\overline {M}}$ we also abstract from the order of precedence of the elements, we get (§1) the cardinal number ${\overline {\overline {M}}}$ of the ordered aggregate $M$, which is, at the same time, the cardinal number of the ordinal type ${\overline {M}}$. From ${\overline {M}}={\overline {N}}$ always follows ${\overline {\overline {M}}}={\overline {\overline {N}}}$, that is to say, ordered aggregates of equal types always have the same power or cardinal number; from the similarity of ordered aggregates follows their equivalence. On the other hand, two aggregates may be equivalent without being similar.

We will use the small letters of the Greek alphabet to denote ordinal types. If a is an ordinal type, we understand by

its corresponding cardinal number.

The ordinal types of finite ordered aggregates offer no special interest. For we easily convince

ourselves that, for one and the same finite cardinal number $\nu$, all simply ordered aggregates are similar to one another, and thus have one and the same type. Thus the finite simple ordinal types are subject to the same laws as the finite cardinal numbers, and it is allowable to use the same signs $1,2,3,...,\nu ,...$ for them, although they are conceptually different from the cardinal numbers. The case is quite different with the transfinite ordinal types; for to one and the same cardinal

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