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Keisler's Order and Full Boolean-Valued Models. (arXiv:1810.05335v1 [math.LO])
来源于:arXiv
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 $\mathbf{M}/\mathcal{U}$
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: $T_0 \trianglelefteq T_1$ 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 $T_0$. 查看全文>>