Uncountable cardinals have the same monadic $\forall _1^1$ positive theory over large sets

Athanassios Tzouvaras¹

¹ Department of Mathematics University of Thessaloniki 541 24 Thessaloniki, Greece

Fundamenta Mathematicae, Tome 181 (2004) no. 2, pp. 125-142.

Résumé

We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form $(\forall X)\phi (X)$ and $(\exists X)\phi (X)$, for $\phi $ positive in $X$ and containing no set-quantifiers, when the set variables range over large ($=$ cofinal) subsets of the cardinals. This strengthens the result of Doner–Mostowski–Tarski [3] that $(\kappa ,\in )$, $(\lambda ,\in )$ are elementarily equivalent when $\kappa $, $\lambda $ are uncountable. It follows that we can consistently postulate that the structures $(2^\kappa ,[2^\kappa ]^{>\kappa },)$, $(2^\lambda ,[2^\lambda ]^{>\lambda },)$ are indistinguishable with respect to $\forall _1^1$ positive sentences. A consequence of this postulate is that $2^\kappa =\kappa ^+$ iff $2^\lambda =\lambda ^+$ for all infinite $\kappa $, $\lambda $. Moreover, if measurable cardinals do not exist, GCH is true.

Détail

DOI : 10.4064/fm181-2-3

Keywords: uncountable cardinals indistinguishable sentences monadic second order language order form forall phi exists phi phi positive containing set quantifiers set variables range large cofinal subsets cardinals strengthens result doner mostowski tarski kappa lambda elementarily equivalent kappa lambda uncountable follows consistently postulate structures kappa kappa kappa lambda lambda lambda indistinguishable respect forall positive sentences consequence postulate kappa kappa lambda lambda infinite kappa lambda moreover measurable cardinals exist gch