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number belong innumerably many different types of simply ordered aggregates, which, in their totality, constitute a particular "class of types" (Typenclasse), Every one of these classes of types is, therefore, determined by the transfinite cardinal number a which is common to all the types belonging to the class. Thus we call it for short the class of types $[{\mathfrak {a}}]$. 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 $[\aleph _{0}]$ which embraces all the types with the least transfinite cardinal number $\aleph _{0}$. From the cardinal number which determines the class of types $[{\mathfrak {a}}]$ we have to distinguish that cardinal number ${\mathfrak {a}}'$ which for its part [499] is determined by the class of types $[{\mathfrak {a}}]$. The latter is the cardinal number which (§1) the class $[{\mathfrak {a}}]$ has, in so far as it represents a well-defined aggregate whose elements are all the types a with the cardinal number ${\mathfrak {a}}$. We will see that ${\mathfrak {a}}'$ is different from ${\mathfrak {a}}$, and indeed always greater than ${\mathfrak {a}}$.

If in an ordered aggregate M all the relations of precedence of its elements are inverted, so that "lower" becomes "higher" and "higher" becomes "lower" everywhere, we again get an ordered aggregate, which we will denote by

and call the "inverse " of $M$. We denote the ordinal type of $^{*}\!M$, if $\alpha ={\overline {M}}$, by