For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

Solution 1:

The proof is really by induction. It goes through the cardinals, and if the claim is false then there is a least one which is a counterexample. Moreover this cardinal is not $\omega$ since for $\omega$ we can check this explicitly. We write $\mu$ for that least counterexample.

Because every two well orders are comparable in the relation "isomorphic to an initial segment", $(\mu,\in)$ and $(\mu\times\mu,<_{cw})$ are comparable. So one is an initial segment of the other, perhaps a proper one. By the fact that $\mu$ is a cardinal and $|\mu|\leq|\mu\times\mu|$ it is impossible that the latter is a proper initial segment of the former, and it is necessary that $\mu$ is an initial segment of $\mu\times\mu$.

But by our assumption that the claim fails at $\mu$, $(\mu,\in)$ is only isomorphic to some proper initial segment. So there is some $\eta<\mu$ such that there is an embedding from $(\mu,\in)$ into $(\eta\times\eta,<_{cw})$ -- this is true because you can always find $\eta$ large enough for that: if $\mu$ is isomorphic to the initial segment below $(\alpha,\beta)$ then taking $\eta=\max\{\alpha,\beta\}+\omega$ we guarantee that $\eta\times\eta$ contains all the information we want. Therefore there is some injection from $\mu$ into $\eta$, which is a contradiction to the fact that $\mu$ is a cardinal and $\eta<\mu$.

Another way of looking at it is to see that an ordinal $\alpha$ is an initial ordinal, or cardinal, if and only if it satisfies the following property:

Every proper initial segment of the order $(\alpha,\in)$ has a strictly smaller cardinality than $\alpha$.

The proof can be given in a positive way instead.

Suppose that the claim is true for all infinite $\lambda<\mu$, it is clear that $(\mu,\in)$ is isomorphic to an initial segment of $(\mu\times\mu,<_{cw})$, by the fact that $\mu$ is an initial ordinal.

It suffices to show that every proper initial segment of $\mu\times\mu$ has cardinality $<\mu$. But if $(\alpha,\beta)$ is any point in $\mu\times\mu$, and $\eta=\max\{\alpha,\beta\}$ then $(\alpha,\beta)\leq_{cw}(\eta,\eta)=\max(\eta+1)\times(\eta+1)$, where the last $\max$ is in the $<_{cw}$ order.

Since $|\eta|=|\eta+1|=\lambda<\mu$, for some $\lambda$, by the induction hypothesis $|\eta\times\eta|=|\lambda\times\lambda|=|\lambda|=\lambda$.

It follows, if so, that every proper initial segment of $\mu\times\mu$ must have cardinality strictly less than $\mu$, and therefore $\mu\times\mu$ is order isomorphic to $\mu$.