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Generic Vopěnka’s principle, remarkable cardinals, and the weak proper forcing axiom. (English) Zbl 1417.03260

Summary: 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 \(B\neq A\) in \(\mathcal {C}\) such that \(B\) elementarily embeds into \(A\) in some set-forcing extension. We show that, for \(n\geq 1\), 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 [J. Bagaria, Arch. Math. Logic 51, No. 3–4, 213–240 (2012; Zbl 1250.03108)]. 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 B. Claverie and R. Schindler [J. Symb. Log. 77, No. 2, 475–498 (2012; Zbl 1250.03111)], 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 \(\kappa \geq \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).

MSC:

03E35 Consistency and independence results
03E55 Large cardinals
03E57 Generic absoluteness and forcing axioms
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