Let $L_{\omega_1}$ be a propositional infinitary language. The subscript $\omega_1$ in $L_{\omega_1}$ indicates that disjunctions and conjunctions of all lengths $<\omega_1$ are allowed. In other words, since $\omega_1$ is the first uncountable ordinal, countable (i.e. finite or denumerable) disjunctions and conjunctions are allowed.
Suppose note I have a sequence of sentences $p_0,p_1,\dots$. Is the notation $\bigvee_{i\in\omega} p_i$ adequate for the countable disjunction of all the $p_i$? Is it equivalent to $\bigvee_{i<\omega_1}p_i$
Note that for ordinals $\alpha, \beta$ we have $\alpha <\beta$ iff $\alpha\in\beta$, and that $\omega_1$ is the smallest noncountable ordinal (having all countable ordinals as elements).
Said that, $\bigvee_{i<\omega_1}p_i$ is already an uncountable disjunction. Moreover, $p_i$ is not defined for $\omega\le i<\omega_1$.
So, this formula fails to be in $L_{\omega_1}$.
Instead, for any countable index set $I$ and formulas $p_i\in L_{\omega_1}\ \ (i\in I)$, we do have $\bigvee_{i\in I} p_i\ \in L_{\omega_1}$.
So, the notation with $I=\omega$ is the adequate one.