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Strong downward Löwenheim–Skolem theorems for stationary logics, I
Archive For Mathematical Logic ( IF 0.3 ) Pub Date : 2020-04-09 , DOI: 10.1007/s00153-020-00730-x
Sakaé Fuchino , André Ottenbreit Maschio Rodrigues , Hiroshi Sakai

This note concerns the model theoretic properties of logics extending the first-order logic with monadic (weak) second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem (SDLS) down to \(<\aleph _2\) for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into four principles. The strongest of these four is shown to be equivalent to the conjunction of CH and the Diagonal Reflection Principle for internally clubness of S. Cox. We show that a further strengthening of this SDLS and its variations follow from the Game Reflection Principle of B. König and its generalizations.



中文翻译:

平稳逻辑的强向下Löwenheim–Skolem定理,I

本说明涉及逻辑的模型理论性质,该逻辑用装备了平稳性量词的单调(弱)二阶变量扩展了一阶逻辑。对于这种逻辑,将向下阶的Löwenheim-Skolem定理(SDLS)降到\(<\ aleph _2 \)的八个变体,并将二阶变量解释为结构的可数子集,可分为四个原理。这四个中最强的一个等效于CH和对角线反射原理对S. Cox的内部俱乐部性的结合。我们表明,根据B.König的博弈反射原理及其概括,可以进一步增强此S​​DLS及其变体。

更新日期:2020-04-09
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