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

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Tzouvaras A.

Fundamenta Mathematicae

2004

tom 181

nr 2

125-142

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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1 Fundamenta Mathematicae

1 Tzouvaras A.

1 2004

Wyniki wyszukiwania

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