My question is about Exercise 3) d) of 4.2 of Introduction to cardinal arithmetic(Holz, Steffens and Weitz):
3) d) Show that $(\omega^{\omega}, \in)$ is not elementary equivalent to $(\aleph_1, \in)$.
I could prove that $(\omega^{\omega}, \in, \times 2)$ is not elementary equivalent to $(\aleph_1, \in, \times 2)$, where $\times 2$ denotes $x \mapsto x\times 2$ (ordinal product), but I can't prove the original problem. Could someone help me? Thanks.
I could always be missing something obvious, but I believe the result is not correct (so either it was miscopied or the authors made a typo). Theorem $6.21$ of Rosenstein's Linear Orderings implies that $$\omega^\omega\cdot \alpha\equiv \omega^\omega\cdot \beta$$ for any two (nonzero) ordinals $\alpha,\beta$. In particular, taking $\alpha=1$ and $\beta=\omega_1$ yields $$\omega^\omega\equiv\omega_1.$$ (I'm identifying an ordinal $\alpha$ with the linear order $(\alpha;\in)$; this is a minor abuse of notation, but is standard practice.)