Regarding some symbols in set theory.

I came across this interesting statement that defines the aleph numbers: $$\aleph_{n+1} = \bigcap \{ x \in \operatorname{On} : | \aleph_n | \lt |x| \}$$ However, I do not understand a couple of symbols. First of all, what does that massive intersection sign mean, and what set is $\operatorname{On}$? Any help would be greatly appreciated.

Solution 1:

$\mathrm{On}$ is presumably the proper class of all ordinals, so $\bigcap\{x \in \mathrm{On} : |\aleph_n| < |x|\}$ is the intersection of all ordinals whose cardinality is strictly greater than that of $\aleph_n$. Since the ordinals are well-ordered with respect to inclusion, this means that $\bigcap\{x \in \mathrm{On} : |\aleph_n| < |x|\}$ is the least ordinal whose cardinality is strictly greater than $|\aleph_n| (= \aleph_n)$, i.e., $\aleph_{n+1}$.

This can be extended to all alephs $\aleph_\alpha$ and may also be used to define $\aleph_\alpha$ for nonzero limit ordinals $\alpha$ as well (namely, for $\alpha$ a nonzero limit ordinal, $\aleph_\alpha = \bigcap\{x \in \mathrm{On} : (\forall \beta < \alpha) \, (|\aleph_\beta| < |x|)\}$).