# $\omega_{1}$-like dense linear ordering **Theorem ([[Hutchinson - Model theory via set theory|Hutchinson]]/Silver).** Every two $\omega_{1}$-like dense linear orderings with first element are $L_{\infty\omega_{1}}$-equivalent. In particular: Let $\eta$ be the order type of the rational numbers. Then $1+\eta\cdot\omega_{1} \equiv_{\infty\omega_{1}}(1+\eta)\cdot\omega_{1}$ ## The orders $\Phi(A)$ >[!info] Definition >With each set $A \subseteq \omega_1$ associate an ordered set $\Phi(A)$ whose order type is $\Sigma_{\alpha<\omega_1}\tau_\alpha$ where > $ > \tau_\alpha=\begin{cases} > 1+\eta & \alpha \in A \\ > \eta & \alpha \notin A > \end{cases} > $ >>[!note] Remark >>$\Phi(A)$ is always an $\omega_1$-like d.l.o., which has a first element iff $0 \in A$. **Theorem ([[Conway - Homogeneous ordered sets|Conway]]).** 1. Every $\omega_1$-like d.l.o. is isomorphic to $\Phi(A)$ for some $A \subseteq \omega_1$. 2. If $A, B \subseteq \omega_1, 0 \in A \cap B$ then $\Phi(A) \cong \Phi(B)$ iff $A \cap C=B \cap C$ for some closed unbounded $C \subseteq \omega_1$.