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

Athanassios Tzouvaras

Fundamenta Mathematicae, Tome 184 (2004), p. 125-142 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $(2^κ,[2^κ]^{>κ},<)$, $(2^λ,[2^λ]^{>λ},<)$ are indistinguishable with respect to ∀₁¹ positive sentences. A consequence of this postulate is that $2^κ = κ⁺$ iff $2^λ = λ⁺$ for all infinite κ, λ. Moreover, if measurable cardinals do not exist, GCH is true.

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Publié le : 2004-01-01

Zbl 1051.03036

EUDML-ID : urn:eudml:doc:283158