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The $\Pi^1_1 \! \! \downarrow$ L\'owenheim-Skolem-Tarski property of Stationary Logic

Authors :
Cox, Sean
Publication Year :
2020

Abstract

Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the L\"owenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $\Pi^1_1$ formulas, which we call the $\Pi^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $\Pi^1_1 \! \! \downarrow$-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.<br />Comment: submitted to RIMS Kokyuroku

Subjects

Subjects :
Mathematics - Logic

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2003.12692
Document Type :
Working Paper