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Exercise I.13.39 in Kunen's Set Theory:

If $\kappa$ is an infinite cardinal and $\alpha = \cup_{n < c}X_n$, where $c < \omega$ and each $\textrm{type}(X_n) < \kappa^\omega$, then $\alpha < \kappa^\omega$.

Using $\textrm{type}(X_n)$ implies that $X_n$ are well-ordered. Does the claim should be modified to: $\textbf{type}(\alpha) < \kappa^\omega$ ?

Should we also assume that the union definition of $\alpha$ is well ordered consistent with the well-ordering of each $X_n$ ?

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Interpreting from context, $\alpha$ is already assumed to be an ordinal. We simply say that if we can write this ordinal as a finite union of sets, each of whom has order type $<\kappa^\omega$, then $\alpha$ is also smaller than $\kappa^\omega$.

Since the $X_n$ are assumed to be subsets of $\alpha$, they are naturally well-ordered.

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