Order Isomorphism between $\omega_1$ and $\omega_1 \times \omega_1$ It is well known that $\omega_1$ and $\omega_1 \times \omega_1$ are order isomorphic when the latter is equipped with canonical well ordering. In fact, it can be explicitly given by $$\phi: \omega_1 \times \omega_1 \to \omega_1 $$ by
$(\alpha,\beta) \mapsto \text{order type of the initial segment generated by } (\alpha,\beta)$.
However, my question comes from an excerpt from M. E. Rudin's text $\textit{Lectures on set theoretic topology}$ where she is showing certain equivalent statements of $\diamondsuit$ which states

there is a one to one function form $\omega_1$ onto $\omega_1 \times \omega_1$ which maps $\alpha$ onto $\alpha \times \alpha$ for each limit $\alpha$

I've tried proving that $\phi$ satisfies this property by inducting over over the limit ordinals, $\Lambda_{\omega_1}$, but can only seems to verify it for $0$.
Any tips or hints? Should  I try to find such a bijection that does not necessarily preserve order?
Edit: From the comments, it is now apparent that this $\phi$ will certainly not work. Then the question now shifts to: what exactly is this bijection which maps $\alpha$ onto $\alpha \times \alpha$ for all limit $\alpha$?
 A: Suppose $\phi : \omega_1 \to \omega_1 \times \omega_1$ is as described by Rudin.  Let $\alpha$ be any countable limit ordinal, and let $\beta = \alpha + \omega$.  Note that $\{\alpha\} \times \alpha$ is contained in $(\beta \times \beta) \setminus (\alpha \times \alpha)$, and therefore it must be contained in the image of $\beta \setminus \alpha$ under $\phi$.  Now let $f : \omega \to (\beta \setminus \alpha)$ be the bijection $f(n) = \alpha + n$.  Then there is a set $A \subseteq \omega$ such that $\phi \circ f \upharpoonright A$ is a bijection from $A$ to $\{\alpha\} \times \alpha$.  So if we let $g : \omega \to A$ be the enumeration of $A$ in increasing order and $h : \omega_1 \times \omega_1 \to \omega_1$ be projection onto the second coordinate, then $h \circ \phi \circ f \circ g$ is a bijection from $\omega$ to $\alpha$.  The upshot of this argument is that any such $\phi$ must encode bijections from $\omega$ to all countable limit ordinals.
Conversely, suppose that for each countable limit ordinal $\alpha$, $\phi_\alpha$ is a bijection from $\omega$ to $\alpha$.  For any limit ordinal $\alpha$, let $\beta = \alpha + \omega$.  Then it is not hard to use $\phi_\alpha$ to define a bijection from $\beta \setminus \alpha$ to $(\beta \times \beta) \setminus (\alpha \times \alpha)$.  Combining these bijections with a bijection from $\omega$ to $\omega \times \omega$ gives the required $\phi$ from $\omega_1$ to $\omega_1 \times \omega_1$.
