# 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

## Access Full Article

top## Abstract

top## How to cite

topAthanassios 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.