发表机构
University of Gothenburg（哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义量词的辖域支配有效性，给出与分支积相等的判据及可数特征新证明，并揭示超滤子与一般量词在等价性上的差异及与可测基数的联系。
AI 中文摘要
设 $P$ 和 $Q$ 是固定论域 $D$ 上的真向上单调一元广义量词。我们研究在所有以 $D$ 为论域的结构 $M$ 上，公式 $M \models Px\\,Qy\\,R(x,y) \rightarrow Qy\\,Px\\,R(x,y)$ 的有效性，该公式称为辖域支配。两个量词的迭代介于其分支积（由矩形生成）与其对偶的分支积之对偶之间。我们给出了与这些边界相等的判据，获得了辖域支配可数特征的新证明，并刻画了内层量词为“至少 $\kappa$ 个”的情形。在广义连续统假设下，Goldberg 定理表明，对于超滤子，辖域支配等价于迭代与分支积相等，而一个在大小为 $\aleph_2$ 的论域上的初等反例表明，对于任意向上单调量词，该等价关系不成立。对于两个量词均为滤子的情形，问题变为集合论问题：反例的存在性与可测基数的存在性等一致。
英文摘要
Let $P$ and $Q$ be proper upward monotone unary generalized quantifiers on a fixed domain $D$. We study the validity, over all structures $M$ with domain $D$, of \[ M \models Px\,Qy\,R(x,y) \rightarrow Qy\,Px\,R(x,y), \] called scope dominance. The iteration of two quantifiers lies between their branching product, which is generated by rectangles, and the dual of the branching product of their duals. We give criteria for equality with these bounds, obtain a new proof of the countable characterization of scope dominance, and characterize the case in which the inner quantifier is "at least $κ$ many." Under the Generalized Continuum Hypothesis, Goldberg's theorem shows that, for ultrafilters, scope dominance is equivalent to equality between iteration and the branching product, whereas an elementary counterexample on a domain of size $\aleph_2$ shows that this equivalence fails for arbitrary upward monotone quantifiers. For the case where both quantifiers are filters, the question becomes set-theoretic: the existence of a counterexample is equiconsistent with the existence of a measurable cardinal.
Comments16 pages
