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

• Autorzy: Athanassios Tzouvaras
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03E10: Ordinal and cardinal numbers
• 03E70: Nonclassical and second-order set theories

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2004
• Tom: 181
• Numer: 2
• Strony: 125-142

Daty

wydano

2004

Twórcy

autor

Athanassios Tzouvaras

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

Identyfikatory

DOI

10.4064/fm181-2-3