Name
Title | ||
|---|---|---|
Generic Vopěnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom. |
Abstract | ||
|---|---|---|
We introduce and study the first-order Generic Vopĕnka's Principle, which states that for every definable proper class of structures $$\\mathcal {C}$$C of the same type, there exist $$B\\ne A$$BźA in $$\\mathcal {C}$$C such that B elementarily embeds into A in some set-forcing extension. We show that, for $$n\\ge 1$$nź1, the Generic Vopĕnka's Principle fragment for $$\\Pi _n$$źn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for $$n\\in \\omega $$nźź, which we introduce here, is a natural generic analogue for the $$C^{(n)}$$C(n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopĕnka'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, $$\\mathrm{wPFA}$$wPFA. The axiom $$\\mathrm{wPFA}$$wPFA states that for every transitive model $$\\mathcal M$$M in the language of set theory with some $$\\omega _1$$ź1-many additional relations, if it is forced by a proper forcing $$\\mathbb P$$P that $$\\mathcal M$$M satisfies some $$\\Sigma _1$$Σ1-property, then V has a transitive model $$\\bar{\\mathcal M}$$M¯, satisfying the same $$\\Sigma _1$$Σ1-property, and in some set-forcing extension there is an elementary embedding from $$\\bar{\\mathcal M}$$M¯ into $$\\mathcal M$$M. This is a weakening of a formulation of $$\\mathrm{PFA}$$PFA due to Claverie and Schindler (J Symb Logic 77(2):475---498, 2012), which asserts that the embedding from $$\\bar{\\mathcal M}$$M¯ to $$\\mathcal M$$M exists in V. We show that $$\\mathrm{wPFA}$$wPFA is equiconsistent with a remarkable cardinal. Furthermore, the axiom $$\\mathrm{wPFA}$$wPFA implies $$\\mathrm{PFA}_{\\aleph _2}$$PFAź2, the Proper Forcing Axiom for antichains of size at most $$\\omega _2$$ź2, but it is consistent with $$\\square _\\kappa $$źź for all $$\\kappa \\ge \\omega _2$$źźź2, and therefore does not imply $$\\mathrm{PFA}_{\\aleph _3}$$PFAź3. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1007/s00153-016-0511-x | Arch. Math. Log. |
Keywords | Field | DocType |
Large cardinals, Vopěnka’s Principle, Generic Vopěnka’s Principle, Remarkable cardinals, Proper Forcing Axiom, 03E35, 03E55, 03E57 | Set theory,Discrete mathematics,Combinatorics,Embedding,Vopěnka's principle,Elementary equivalence,Axiom,Cardinal number,Omega,Proper forcing axiom,Mathematics | Journal |
Volume | Issue | ISSN |
56 | 1-2 | 1432-0665 |
Citations | PageRank | References |
4 | 0.74 | 7 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Joan Bagaria | 1 | 63 | 13.15 |
| Victoria Gitman | 2 | 13 | 6.59 |
| ralf schindler | 3 | 55 | 15.49 |