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

Fundamenta Mathematicae (2004)

• Volume: 181, Issue: 2, page 125-142
• ISSN: 0016-2736

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## Abstract

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

## How to cite

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Athanassios Tzouvaras. "Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets." Fundamenta Mathematicae 181.2 (2004): 125-142. <http://eudml.org/doc/283158>.

@article{AthanassiosTzouvaras2004,
abstract = {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.},
author = {Athanassios Tzouvaras},
journal = {Fundamenta Mathematicae},
keywords = {monadic second-order logic of order; formulas with first-order quantification; positive formulas with only one set variable; expressiveness over transfinite cardinals},
language = {eng},
number = {2},
pages = {125-142},
title = {Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets},
url = {http://eudml.org/doc/283158},
volume = {181},
year = {2004},
}

TY - JOUR
AU - Athanassios Tzouvaras
TI - Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets
JO - Fundamenta Mathematicae
PY - 2004
VL - 181
IS - 2
SP - 125
EP - 142
AB - 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.
LA - eng
KW - monadic second-order logic of order; formulas with first-order quantification; positive formulas with only one set variable; expressiveness over transfinite cardinals
UR - http://eudml.org/doc/283158
ER -

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