For infinite cardinals $\kappa$, we have $\kappa \otimes \kappa = \kappa$.

Question

I am aware that other questions are quite similar to this; however, it seems like the other questions regarding the same statement are looking at proofs that seem somewhat different from the one I am interested in, so here goes:

On page $29$ in Kunen's "Set Theory Introduction to Independence Proofs", he wants to prove that for infinite cardinals $\kappa$, we have $\kappa \otimes \kappa = \kappa$.

I'll provide the proof in full (paraphrased):

We prove this by transfinite induction on $\kappa$. Assume this holds for smaller cardinals. Then for $\alpha < \kappa$, we have \begin{align*} \alpha \otimes \alpha &= |\alpha \times \alpha|\\ &= |\alpha| \otimes |\alpha|\\ &< \kappa \end{align*} (applying lemma 10.10 when $\alpha$ is finite, which says that for $m,n \in \omega: m \otimes n < \omega$). Define a well-ordering $\triangleleft$ on $\kappa \times \kappa$ by $$ \langle \alpha,\beta \rangle \ \triangleleft \ \langle \gamma,\delta \rangle \iff \text{max}(\alpha,\beta) < \text{max}(\gamma,\delta) \lor \left[\text{max}(\alpha,\beta) = \text{max}(\gamma,\delta) \land \langle \alpha,\beta \rangle \ \text{precedes} \ \langle \gamma,\delta \rangle \ \text{lexicographically}\right]. $$ Each $\langle \alpha,\beta \rangle \in \kappa \times \kappa$ has no more than $|(\text{max}(\alpha,\beta)+1) \times (\text{max}(\alpha,\beta)+1)| < \kappa$ predecessors in $\triangleleft$, so $\text{type}(\kappa \times \kappa, \triangleleft) \leq \kappa$, whence $|\kappa \times \kappa| \leq \kappa$. Since clearly $|\kappa \times \kappa| \geq |\kappa| = \kappa$, we have $$ |\kappa \times \kappa| = |\kappa| = \kappa.$$

Now, I am having a really hard time trying to understand even the structure of this proof. For example, as a friend pointed out, why is he taking account of finite cardinals $\alpha$ in the proof? Is it because the induction is really on showing that for all $\alpha < \kappa$ we have $|\alpha \times \alpha| \leq \kappa$, and the induction is not on the property $\kappa \otimes \kappa = \kappa$. The first induction makes sense since if he was really doing induction on $\kappa \otimes \kappa = \kappa$ for infinite cardinals, the base case would be $\omega$ (and we could disregard finite $\alpha$). I have looked at this proof for quite a while but I am still having trouble understanding the logical structure of the proof.

I would appreciate anyone who could elucidate this clearly for me.

Answer 1

Among other reasons, he's accounting for finite cardinals in order to apply the proof to $\aleph_0 \otimes \aleph_0$.

The structure of the proof is just that for any $\alpha \lt \kappa$, we know by the induction hypothesis that $\vert \alpha \times \alpha \vert \lt \kappa$. Using this knowledge, we create a well ordering of $\kappa \times \kappa$ such that any ordered pair $(\alpha, \beta) \in \kappa \times \kappa$ has fewer than $\kappa$ predecessors. That well order tells us that $\vert \kappa \times \kappa \vert \leq \kappa$.