Supremum of ordinal sequence under addition

Question

I am trying to prove statement (6.1) in Chapter 6 of Introduction to Set Theory by Hrbacek and Jech. According to this if $\alpha$ is any ordinal, $\{\beta_\nu\}$ is a transfinite sequence indexed by nonzero limit ordinal $\gamma$, and $\beta = \sup_{\nu < \gamma} \beta_\nu$ then $$ \alpha + \beta = \sup_{\nu < \gamma} (\alpha + \beta_\nu) \,. $$ The text says that this follows immediately from the recursive definition of ordinal addition (and doesn't bother to provide a proof) but, in trying to be adequately rigorous, the proof I came up with is hardly trivial and had to involve several Lemmas. Am I missing something simple here or were the authors stretching it when they said it follows immediately from the definition?

Answer 1

After discussion it seems that there is no very short, trivial proof of this. However, the other statements of (6.1) in the text also assert this "continuity" property for ordinal multiplication and exponentiation. That is $$ \alpha \cdot \beta = \sup_{\nu < \gamma} \alpha \cdot \beta_\nu $$ and $$ \alpha^\beta = \sup_{\nu < \gamma} \alpha^{\beta_\nu} \,. $$ These properties can all be proved similarly with the use of two lemmas that apply to general monotonically increasing ordinal functions.

Lemma 1: If $A$ is a nonempty set of ordinals, $\alpha$ is the greatest element of $A$, and $f$ a monotonically increasing function defined on $A$ then $f(\alpha) = \sup_{\beta \in A} f(\beta)$.

Lemma 2: If $A$ and $B$ are sets of ordinals where $\sup{A} = \sup{B}$ but neither have a greatest element and $f$ is a monotonically increasing function defined on $A \cup B$ then $\sup_{\alpha \in A} f(\alpha) = \sup_{\beta \in B} f(\beta)$.

Neither of these are particularly difficult to prove so I won't give proofs here. We then define the set $A$ as the range of the transfinite sequence $\{\beta_\nu\}$ so that $\beta = \sup{A}$. For each operation (addition, multiplication, and exponentiation) we then need only show that they are monotonically increasing. Then we apply Lemma 1 to the case when $\beta \in A$ (so that it is the greatest element of $A$) and Lemma 2 to the case when $\beta \notin A$ (so that $A$ has no greatest element). In the latter case we must apply the definition of the operation when $\beta$ is a limit ordinal before applying Lemma 2.

Note that exponentiation is only monotonic when $\alpha > 0$ since $0^1 = 0 < 1 = 0^0$. Hence the continuity result only holds for $\alpha > 0$, which is actually not mentioned in the book.