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Uncountable cardinals have the same monadic ∀11positive theory over large sets

2004
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Fundamenta Mathematicae
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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)φ(X) and (∃X)φ(X), for φ 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 (κ, ∈), (λ, ∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures (2 κ , [2 κ ] >κ

doi:10.4064/fm181-2-3
fatcat:2zsb5csljzczpen7ttizdgsoji