Let $(X_\alpha)$ be a nested ordinal-indexed sequence of sets, i.e. if $\alpha\leq\beta$ then $X_\alpha\subseteq X_\beta$ for all ordinals $\alpha$ and $\beta$, and suppose that there is a cardinal $\kappa$ such that $|X_\alpha|\leq\kappa$ for all ordinals $\alpha$. My question is, what can we say about the cardinality of $\cup_\alpha X_\alpha$?
First of all $\cup_\alpha X_\alpha$ is a set as opposed to a proper class? Second of all, can we put some kind of bound on the cardinality of $\cup_\alpha X_\alpha$?
This came up in the course of writing up a proof of the downward Lowenheim-Skolem theorem, by the way.
Best Answer
Since the chain is well-ordered, it is necessarily stabilized by some $\gamma<\kappa^+$, otherwise choosing some $x_\alpha\in X_{\alpha+1}\setminus X_\alpha$ for $\alpha<\kappa^+$, would define an injection from $\kappa^+$ into $X_{\kappa^+}$.
So indeed the union is a set, and its cardinality is at most $\kappa$. And easily, for any cardinal $\lambda\leq\kappa$, the union could have cardinality $\lambda$ (which means that it may have stabilized before $\kappa$).