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ence. If ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle P}$, ${\displaystyle ...}$ are aggregates which have no common elements, ${\displaystyle M'}$, ${\displaystyle N'}$, ${\displaystyle P'}$, ${\displaystyle ...}$ are also aggregates with the same property, and if

then we always have

If for two aggregates ${\displaystyle M}$ and ${\displaystyle N}$ with the cardinal numbers ${\displaystyle {\mathfrak {a}}={\overline {\overline {M}}}}$ and ${\displaystyle {\mathfrak {b}}={\overline {\overline {N}}}}$, both the conditions:

(a) There is no part of ${\displaystyle M}$ which is equivalent to ${\displaystyle N}$,
(b) There is a part ${\displaystyle N_{1}}$ of ${\displaystyle N}$, such that ${\displaystyle N_{1}\sim M}$,

are fulfilled, it is obvious that these conditions still hold if in them ${\displaystyle M}$ and ${\displaystyle N}$ are replaced by two equivalent aggregates ${\displaystyle M'}$ and ${\displaystyle N'}$. Thus they express a definite relation of the cardinal numbers ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ to one another.

[484] Further, the equivalence of ${\displaystyle M}$ and ${\displaystyle N}$, and thus the equality of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$, is excluded; for if we had ${\displaystyle M\sim N}$, we would have, because ${\displaystyle N_{1}\sim M}$, the equivalence ${\displaystyle N_{1}\sim N}$, and then, because ${\displaystyle M\sim N}$, there would exist a part ${\displaystyle M_{1}}$ of ${\displaystyle M}$ such that ${\displaystyle M_{1}\sim M,}$ and therefore we should have ${\displaystyle M_{1}\sim N}$; and this contradicts the condition (a).

Thirdly, the relation of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ is such that it makes impossible the same relation of ${\displaystyle {\mathfrak {b}}}$ and ${\displaystyle {\mathfrak {a}}}$; for if