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

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

(1)

${\displaystyle {\mathfrak {a}}<{\mathfrak {b}}}$ or ${\displaystyle {\mathfrak {b}}>{\mathfrak {a}}.}$

We can easily prove that,

(2)

if ${\displaystyle {\mathfrak {a}}<{\mathfrak {b}}}$ and ${\displaystyle {\mathfrak {b}}<{\mathfrak {c}}}$, then we always have ${\displaystyle {\mathfrak {a}}<{\mathfrak {c}}}$.

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

We have seen that, of the three relations

each one excludes the two others. On the other hand, the theorem that, with any two cardinal numbers ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$, 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 ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ are any two cardinal numbers, then either ${\displaystyle {\mathfrak {a}}={\mathfrak {b}}}$ or ${\displaystyle {\mathfrak {a}}<{\mathfrak {b}}}$ or ${\displaystyle {\mathfrak {a}}>{\mathfrak {b}}}$.

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