I'm trying to show that ordinal addition is associative, i.e. for ordinals $\alpha, \beta$ and $\gamma$ we have
$$(\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)$$
My idea is to show that the identity map is a bijection between the set on the left and the one on the right. Then it will trivially be an order isomorphism and as every ordinal is order isomorphic to itself only, then the statement will follow.
The identity map is obviously injective, but I'm having troubles verifying that it is surjective. Also, as easy as this may seem, I can't show that this map is well-defined, that is that it maps the LHS to the RHS.
I had a look at this answer, but to no success - Ordinal addition is associative
Any help is appreciated!
This should be tagged elementary set-theory.
First show that your definition of ordinal sum is equivalent to the following: $\alpha + \beta$ is the lexicographic order on $\{0\} \times \alpha \cup \{1\} \times \beta$ (a copy of $\alpha$ followed by a copy of $\beta$). Now associativity is obvious.