Hierarchies of (Virtual) Resurrection Axioms

Abstract

I analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings (the emphasis being on the subcomplete forcings). I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies and the hierarchies of bounded and weak bounded forcing axioms.

Subcomplete Forcing, Trees, and Generic Absoluteness

“…Since the forcing is subcomplete in L, it does not add reals, and hence preserves CH. In a sense, in the context of BSCFA, it is natural to assume CH, since it holds in the "natural" models, and since CH is implied by natural strengthenings of BSCFA, such as the resurrection axiom or the maximality principle for subcomplete forcing, see [13], [4].…”

Combining Resurrection and Maximality

“…Hamkins and Johnstone [HJ14a, Section 6] combine the resurrection axiom with forcing axioms, like PFA for example, and show that they hold after a forcing iteration. Fuchs explores combinations of maximality principles for closed forcing notions [Fuc08] combined with those for collapse and directed closed forcing notions [Fuc09] and hierarchies of resurrection axioms with an emphasis on subcomplete forcing [Fuc18a]. Trang and Ikegami studied classes of maximality principles combined with forcing axioms [IT18].…”

Subcomplete forcing principles and definable well‐orders

“…Other previous research focused on forcing principles for subcomplete forcing other than the traditional axioms, such as resurrection axioms (cf. ) or maximality principles (cf. ).…”

“…In the paper [11], I determined the effects of SCFA on the failure of weak variants of the Todorčević square principles, and again, the situation turned out to be very similar to that of MM -the only difference stemming from the fact that SCFA does not imply the failure of CH. Other previous research focused on forcing principles for subcomplete forcing other than the traditional axioms, such as resurrection axioms (see [8]) or maximality principles (see [24]).…”