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

• Published in: Archive for mathematical logic Vol. 56; no. 1-2; pp. 1 - 20
• Main Authors: Bagaria, Joan, Gitman, Victoria, Schindler, Ralf
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2017; Springer Nature B.V
• Subjects: Algebra; Arches; Embedding; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Set theory; Remarkable cardinals; Generic Vopěnka’s Principle; 03E35; 03E57; Large cardinals; Vopěnka’s Principle; Proper Forcing Axiom; 03E55
• ISSN: 0933-5846; 1432-0665
• DOI: 10.1007/s00153-016-0511-x

We introduce and study the first-order Generic Vopěnka’s Principle , which states that for every definable proper class of structures C of the same type, there exist B ≠ A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1 , the Generic Vopěnka’s Principle fragment for Π n -definable classes is equiconsistent with a proper class of n -remarkable cardinals. The n -remarkable cardinals hierarchy for n ∈ ω , 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 , wPFA . The axiom wPFA states that for every transitive model M in the language of set theory with some ω 1 -many additional relations, if it is forced by a proper forcing P that M satisfies some Σ 1 -property, then V has a transitive model M ¯ , satisfying the same Σ 1 -property, and in some set-forcing extension there is an elementary embedding from M ¯ into M . This is a weakening of a formulation of PFA due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012 ), which asserts that the embedding from M ¯ to M exists in V . We show that wPFA is equiconsistent with a remarkable cardinal. Furthermore, the axiom wPFA implies PFA ℵ 2 , the Proper Forcing Axiom for antichains of size at most ω 2 , but it is consistent with □ κ for all κ ≥ ω 2 , and therefore does not imply PFA ℵ 3 .