In Tao's book on random matrix theory, we have Lemma 1.1.3 (see here), but I'm having trouble parsing exactly what's meant by (for example) "$E_\alpha$ is a family of events of polynomial cardinality."
For fixed $\alpha$ in some index set $I$, it's mentioned on the previous page that the event $E_\alpha$ should depend on some parameter $n$, and $E_\alpha(n)$ has the appropriate asymptotic behavior as defined before the lemma. But I'm a little confused about what it means to say that the family $\{E_\alpha\}$ has cardinality $O(n^{O(1)})$ here. At first I thought that maybe we mean that there's a function $f: \mathbb{N} \rightarrow 2^I$ such that every $\alpha$ belongs to $f(n)$ for some $n$ and the cardinality of $f(n)$ is $O(n^{O(1)})$, but this seems to be equivalent to $I$ being countable. So I suspect I'm not reading this correctly.
Maybe we mean that the cardinality of the set $\cup_\alpha \{E(n, \alpha)\}$ is $O(n^{O(1)})$? But this seems like an unnatural thing to consider in this context. Any thoughts?