## Abstract

We study various classes of maximality principles, \(\mathrm {MP}(\kappa ,\Gamma )\), introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where \(\Gamma \) defines a class of forcing posets and \(\kappa \) is an infinite cardinal. We explore the consistency strength and the relationship of \(\textsf {MP}(\kappa ,\Gamma )\) with various forcing axioms when \(\kappa \in \{\omega ,\omega _1\}\). In particular, we give a characterization of bounded forcing axioms for a class of forcings \(\Gamma \) in terms of maximality principles MP\((\omega _1,\Gamma )\) for \(\Sigma _1\) formulas. A significant part of the paper is devoted to studying the principle MP\((\kappa ,\Gamma )\) where \(\kappa \in \{\omega ,\omega _1\}\) and \(\Gamma \) defines the class of stationary set preserving forcings. We show that MP\((\kappa ,\Gamma )\) has high consistency strength; on the other hand, if \(\Gamma \) defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP\((\kappa ,\Gamma )\) is consistent relative to \(V=L\).

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## References

Audrito, G., Viale, M.: Absoluteness via Resurrection. arXiv:1404.2111

Bagaria, J.: Bounded forcing axioms as principles of generic absoluteness. Arch. Math. Log.

**39**(6), 393–401 (2000)Claverie, B., Schindler, R.: Woodin’s axiom (*), bounded forcing axioms, and precipitous ideals on \(\omega _1\). J. Symb. Log.

**77**(2), 475–498 (2012)Cummings, J.: Iterated forcing and elementary embeddings. In: Handbook of Set Theory, pp. 775–883 (2010)

Fuchs, G., Hamkins, J.D., Reitz, J.: Set-theoretic geology. Ann. Pure Appl. Log.

**166**(4), 464–501 (2015)Hamkins, J.D., Woodin, W.H.: The necessary maximality principle for ccc forcing is equiconsistent with a weakly compact cardinal. Math. Log. Q.

**51**(5), 493–498 (2005)Hamkins, J.D.: A simple maximality principle. J. Symb. Log.

**68**(2), 527–550 (2003)Hamkins, J.D., Johnstone, T.A.: Resurrection axioms and uplifting cardinals. Arch. Math. Log.

**53**(3–4), 463–485 (2014)Larson, P.B.: Martin’s maximum and definability in \(H(\aleph _2)\). Ann. Pure Appl. Log.

**156**(1), 110–122 (2008)Moore, J.T.: Set mapping reflection. J. Math. Log.

**5**(01), 87–97 (2005)Schindler, R., Steel, J.R.: The Core Model Induction. http://math.berkeley.edu/~steel

Schindler, R.: Woodin’s Axiom (*), or Martin’s Maximum, or Both. http://wwwmath.uni-muenster.de/u/rds

Schindler, R.: Coding in K by reasonable forcing. Trans. Am. Math. Soc.

**353**, 479–489 (2001)Schindler, R.: Semi-proper forcing, remarkable cardinals, and bounded Martin’s maximum. Math. Log. Q.

**50**(6), 527–532 (2004)Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)

Todorcevic, S.: Generic absoluteness and the continuum. Math. Res. Lett.

**9**(4), 465–472 (2002)Tsaprounis, K.: On resurrection axioms. J. Symb. Log.

**80**(2), 587–608 (2015)Usuba, T.: The downward directed grounds hypothesis and very large cardinals (submitted for publication)

Viale, M.: Category forcings, \( {MM}^{+++}\), and generic absoluteness for the theory of strong forcing axioms. J. Am. Math. Soc.

**29**(3), 675–728 (2016)Viale, M.: Martin’s maximum revisited. Arch. Math. Log.

**55**(1–2), 295–317 (2016)Woodin, W.H.: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Volume 1 de Gruyter Series in Logic and Its Applications. Walter de Gruyter & Co., Berlin (2010)

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Ikegami, D., Trang, N. On a class of maximality principles.
*Arch. Math. Logic* **57**, 713–725 (2018). https://doi.org/10.1007/s00153-017-0603-2

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DOI: https://doi.org/10.1007/s00153-017-0603-2