Keisler's Order and Full Boolean-Valued Models

We prove a compactness theorem for full Boolean-valued models. As an application, we show that if $T$ is a complete countable theory and $\mathcal{B}$ is a complete Boolean algebra, then ${\lambda }^{+}$-saturated $\mathcal{B}$-valued models of $T$ exist. Moreover, if $\mathcal{U}$ is an ultrafilter on $T$ and $\mathbf{M}$ is a ${\lambda }^{+}$-saturated $\mathcal{B}$-valued model of $T$, then whether or not $\mathbf{M}\mathrm{/}\mathcal{U}$ is ${\lambda }^{+}$-saturated just depends on $\mathcal{U}$ and $T$; we say that $\mathcal{U}$ ${\lambda }^{+}$-saturates $T$ in this case. We show that Keisler’s order can be formulated as follows: $T_0⊴T_1$ if and only if for every cardinal $\lambda$, for every complete Boolean algebra $\mathcal{B}$ with the ${\lambda }^{+}$-c.c., and for every ultrafilter $\mathcal{U}$ on $\mathcal{B}$, if $\mathcal{U}$ ${\lambda }^{+}$-saturates $T_1$, then $\mathcal{U}$ ${\lambda }^{+}$-saturates $T_0$.