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.



Why is it important to have disjoint sets in a union for the union to make sense w.r.t the order types?
You didn't say so, but you are speaking of ordinals here and ordinal exponentiation.


The problem is that if you take the union of sets not in order, then you can't control the order type. Let me give an easy example to illustrate the point. Suppose that An for each n < ω consists of a single ordinal. If the ordinal of An is below the ordinal of Am whenever n < m, then the union set Un An will clearly have order type ω. But if I drop that requirement, then I can get any countable ordinal at all! That is, every set is the uion of its singletons. In particular, if α is a countable ordinal, then α = U { {β} | β < α} is the union of countably many one point orders. Indeed, if you go to the non-ordinal context, then every countable order type, such as Q, is the union of countably many singletons. 


A similar problem arises in your specific question. You can't guarrantee that the union of the Anε has small order type, if you allow them to get mixed up all together.


