Keisler's Order and Full Boolean-Valued Models

2018/10/12 by Ulrich, Douglas
#03C55 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1810.05335

Abstract

We prove a compactness theorem for full Boolean-valued models. As an application, we show that if T is a complete countable theory and B is a complete Boolean algebra, then λ⁺-saturated B-valued models of T exist. Moreover, if U is an ultrafilter on T and M is a λ⁺-saturated B-valued model of T, then whether or not M/U is λ⁺-saturated just depends on U and T; we say that U λ⁺-saturates T in this case. We show that Keisler's order can be formulated as follows: T₀ \trianglelefteq T₁ if and only if for every cardinal λ, for every complete Boolean algebra B with the λ⁺-c.c., and for every ultrafilter U on B, if U λ⁺-saturates T₁, then U λ⁺-saturates T₀.

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