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in (a) and (b) the parts played by $M$ and $N$ are interchanged, two conditions arise which are contradictory to the former ones.

We express the relation of ${\mathfrak {a}}$ to ${\mathfrak {b}}$ characterized by (a) and (b) by saying: ${\mathfrak {a}}$ is "less" than ${\mathfrak {b}}$ or ${\mathfrak {b}}$ is "greater" than ${\mathfrak {a}}$; in signs

We can easily prove that,

Similarly, from the definition, it follows at once that, if $P_{1}$ is part of an aggregate $P$, from ${\mathfrak {a}}<{\overline {\overline {P_{1}}}}$ follows ${\mathfrak {a}}<{\overline {\overline {P}}}$ and from ${\overline {\overline {P}}}<{\mathfrak {b}}$ follows ${\overline {\overline {P_{1}}}}<{\mathfrak {b}}$.

We have seen that, of the three relations

${\mathfrak {a}}={\mathfrak {b}}$, ${\mathfrak {a}}<{\mathfrak {b}}$, ${\mathfrak {b}}<{\mathfrak {a}}$,

each one excludes the two others. On the other hand, the theorem that, with any two cardinal numbers ${\mathfrak {a}}$ and ${\mathfrak {b}}$, one of those three relations must necessarily be realized, is by no means self-evident and can hardly be proved at this stage.

Not until later, when we shall have gained a survey over the ascending sequence of the transfinite cardinal numbers and an insight into their connexion, will result the truth of the theorem:

A. If ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are any two cardinal numbers, then either ${\mathfrak {a}}={\mathfrak {b}}$ or ${\mathfrak {a}}<{\mathfrak {b}}$ or ${\mathfrak {a}}>{\mathfrak {b}}$.

From this theorem the following theorems, of which, however, we will here make no use, can be very simply derived: