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Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

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We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).

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Correspondence to Victoria Gitman.

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Parts of this research were done while all three authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in the programme “Mathematical, Foundational and Computational Aspects of the Higher Infinite” (HIF) funded by EPSRC Grant EP/K032208/1 in September 2015. The first author would like to thank the support provided by a Simons Foundation fellowship while at the INI. Other parts of this research were done while the third author was visiting the School of Mathematics of the IPM, Tehran, Iran, in October 2015; he would like to thank his hosts for their exceptional hospitality. The research work of the first author was partially supported by the Spanish Government under Grant MTM2014-59178-P, and by the Generalitat de Catalunya (Catalan Government) under Grant SGR 437-2014.

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Bagaria, J., Gitman, V. & Schindler, R. Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic 56, 1–20 (2017).

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