If $\alpha,\beta$ are at most countable, so are $\alpha+\beta,\alpha\cdot\beta,\alpha^\beta$

Question

If $\alpha$ and $\beta$ are at most countable ordinals, prove the following are at most countable:

• $\alpha + \beta$
• $\alpha \cdot \beta$
• $\alpha^{\beta}$

Answer 1

HINT: There are two possible ways, depending on how you defined the operations.

1. If you have defined all these operations by transfinite recursion. Prove by induction on $\beta$ for addition; use that and the definition of multiplication to prove this by induction on $\beta$ again; and use that fact and the definition of exponentiation to prove - again - with induction on $\beta$.

2. If you have defined all these operations by defining a well-order on some set generated from $\alpha$ and $\beta$. Note that addition is an order of the disjoint union; multiplication is an order of the product; and exponentiation is an order of the finitely non-zero functions from $\beta$ to $\alpha$.

   In every one of these cases one can show that starting with countable sets the result is countable (that is, disjoint union of countable sets is countable; product of two countable sets is countable; the set of all finite sequences of a countable set is countable).