set theory - Why is it important to have disjoint sets in a union for the union to make sense w.r.t the order types?

This question has been bugging me for quite some time now.

Say we have some $beta$ smaller than some $gamma$ and a sequence

$beta$_$epsilon$ : $epsilon$ smaller than cf($beta$) cofinal in $beta$ and say

we have some sets $A$_n^$epsilon$ and each of these $A$_n^$epsilon$ has order type less than $gamma$^$n$.

Now $forall n$ in $omega$ let $B$_n= $cup$ $A$_n^$epsilon$ for all $epsilon$ < $gamma$ and suppose in the end I can write $beta$ as the union of all the $B$_n (but that is not really my problem here)

Why can I deduce that $B$_n has order type less than $gamma$^$n+1$only if all my sets $A$_n^$epsilon$ are disjoint and do not overlap?

(since we have a union of less then $gamma$ sets each of which is of order type less than $gamma$^$n$)

Why can't I still guarantee that the $B$_n will still have order type $gamma$^$n+1$ if all the $A$_n^$epsilon$ are not disjoint?

I know that I need to take the $A$_n^$epsilon$ to be [$epsilon$,$epsilon+1$) so that they are disjoint.

But why does everything in the union have to be in order?

I hope I conveyed my question clearly. Thanks in advance for any help.

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