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

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## Abstract

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 \(\mathrm{PFA}_{\aleph _3}\).

## Keywords

Large cardinals Vopěnka’s Principle Generic Vopěnka’s Principle Remarkable cardinals Proper Forcing Axiom## Mathematics Subject Classification

03E35 03E55 03E57## Preview

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## References

- 1.Bagaria, J.: \(C^{(n)}\)-cardinals. Arch. Math. Logic
**51**(3–4), 213–240 (2012)MathSciNetCrossRefMATHGoogle Scholar - 2.Claverie, B., Schindler, R.: Woodin’s axiom \((\ast )\), bounded forcing axioms, and precipitous ideals on \(\omega _1\). J. Symb. Logic
**77**(2), 475–498 (2012)CrossRefMATHGoogle Scholar - 3.Cheng, Y., Gitman, V.: Indestructibility properties of remarkable cardinals. Arch. Math. Logic
**54**(7), 961–984 (2015)Google Scholar - 4.Cheng, Y., Schindler, R.: Harrington’s principle in higher order arithmetic. J. Symb. Logic
**80**(2), 477–489 (2015)MathSciNetCrossRefMATHGoogle Scholar - 5.Gale, D., Stewart, F.M.: Infinite games with perfect information. Ann. Math. Stud.
**28**, 245–266 (1953)MathSciNetMATHGoogle Scholar - 6.Gitman, V., Schindler, R.: Virtual large cardinals. In preparationGoogle Scholar
- 7.Gitman, V., Welch, P.D.: Ramsey-like cardinals II. J. Symb. Logic
**76**(2), 541–560 (2011)MathSciNetCrossRefMATHGoogle Scholar - 8.Hamkins, J.D.: The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme. Manuscript under reviewGoogle Scholar
- 9.Kunen, K.: Elementary embeddings and infinitary combinatorics. J. Symb. Logic
**36**, 407–413 (1971)MathSciNetCrossRefMATHGoogle Scholar - 10.Magidor, M.: On the role of supercompact and extendible cardinals in logic. Isr. J. Math.
**10**, 147–157 (1971)MathSciNetCrossRefMATHGoogle Scholar - 11.Schindler, R.D.: Proper forcing and remarkable cardinals. Bull. Symb. Logic
**6**(2), 176–184 (2000)MathSciNetCrossRefMATHGoogle Scholar - 12.Schindler, R.: Remarkable cardinals. In: Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch, Series: Tributes. College Publications, London, GB (2014)Google Scholar
- 13.Schindler, R.: Set theory. Universitext. Springer, Cham (2014). Exploring independence and truthGoogle Scholar
- 14.Todorčević, S.: A note on the proper forcing axiom. In: Axiomatic Set Theory (Boulder, CO, 1983), vol. 31 of Contemp. Math., pp. 209–218. Am. Math. Soc., Providence, RI (1984)Google Scholar
- 15.Todorčević, S.: Localized reflection and fragments of PFA. In: Set Theory (Piscataway, NJ, 1999), vol. 58 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 135–148. Am. Math. Soc., Providence, RI (2002)Google Scholar