# Proving that if $\kappa$ is a limit ordinal, then $\alpha+\kappa=\bigcup_{\gamma\in\kappa}\left(\alpha+\gamma\right)$

Let $$\alpha$$ be an ordinal and let $$\kappa$$ be a limit ordinal. I want to prove that $$\alpha+\kappa=\bigcup_{\gamma\in\kappa}\left(\alpha+\gamma\right).$$ Here I define the sum of ordinals not using recursion. I define the sum of two ordinals $$\alpha$$ and $$\beta$$ as follows. Choose well-ordered sets $$A$$ and $$B$$ which are isomorphic to $$\alpha$$ and $$\beta$$, respectively, take their union $$A\cup B$$ and define a well-order on it by making every element of $$A$$ less than every element of $$B$$. Then the unique ordinal which is isomorphic to $$A\cup B$$ is $$\alpha+\beta$$. I want to use this direct definition to prove the question, no other results. I am having trouble with it because it is taking the union of well ordered sets makes it very complicated.

The $$\bigcup$$ is nested union only adding elements that are larger than everything we have before, so there is nothing complicated about this union.

Both LHS and RHS are partial order on $$\alpha\amalg\kappa$$, and it is clear that they are well-ordering on the subset of elements comparable with $$\alpha$$.

LHS is declaring everything in the $$\alpha$$ copy is less than in $$\kappa$$ copy.

RHS is declaring everything in the $$\alpha$$ copy is less than everything inside an element in the $$\kappa$$ copy (which is what the $$\gamma\in\kappa$$ is doing). But $$\kappa$$ being a limit ordinal means $$\gamma\in\kappa\Rightarrow\gamma\in\gamma+1\in\kappa$$, and so we have each $$\gamma\in\kappa$$ is also declared larger than everything in $$\alpha$$. So LHS and RHS defines the same ordering on $$\alpha\amalg\kappa$$.

For each proper initial segment $$C$$ of $$B$$ we have a unique order isomorphism between $$C$$ and an ordinal $$\gamma \in \kappa$$.

This gives us a unique order isomorphism $$f_C : A \cup C \to \alpha + \gamma$$, and by uniqueness all of these isomorphisms agree on their common domain.

So we can take the union $$f = \bigcup f_C$$, which is an order isomorphism between $$A \cup B$$ and $$\bigcup_{\gamma \in \kappa}(\alpha + \gamma)$$.