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

Sean Cox

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arXiv · 2003.12692

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

Sean Cox

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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.

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BibTeX RIS

Sean Cox. 2020-03-28. The $\Pi^1_1 \! \! \downarrow$ L\"owenheim-Skolem-Tarski property of Stationary Logic. https://arxiv.org/abs/2003.12692

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