Uncountable cardinals have the same monadic $\forall _1^1$ positive theory over large sets
Volume 181 / 2004
Abstract
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.
