We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

We study Σ1-definable sets in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1-definable, the set of all stationary subsets of ω1 is not Σ1-definable and the complement of every Σ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1-definable well-ordering (...) of H and the existence of a Δ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1-definable subsets of κκ, in the case where κ itself has certain large cardinal properties. (shrink)

We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...) example of a so-called boldface bounded forcing axiom implying 20=2. (shrink)

We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{*}$ and $A-{\mathsf{BMM}}^{*,++}$, and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{*,++}$ holds true for every set of reals $A$ in $L$, then Woodin’s axiom $$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{*,++}$ for every $A$ which is universally Baire.

Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...) optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin’s Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from ${H_{\omega_2}}$ implies that every function from ω 1 into ω 1 is bounded on a club by a canonical function. (shrink)

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.

We introduce the Bounded Axiom A Forcing Axiom . It turns out that it is equiconsistent with the existence of a regular ∑2-correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom.

We show that in a σ‐closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of J. T. Moore.

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any suborder of the reals of size (...) ω 1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of S κ + ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω 2 ). (shrink)

It is shown that the boldface maximality principle for subcomplete forcing,, together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of definable without parameters. The same conclusion follows from, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that does not exist, for some, implies the existence of a well‐ordering of which is Δ1‐definable without parameters, and ‐definable (...) using a subset of ω1 as a parameter. This well‐order is in. Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed. (shrink)

We explore the consistency strength of Σ 3 1 and Σ 4 1 absoluteness, for a variety of forcing notions.

We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our investigation (...) of this helical structure, we have new characterizations of extendible cardinals and of Vopěnkan cardinals in terms of elementary embeddings from the universe V, a new large large cardinal property which looks like Shelahness properly between Vopěnkaness and almost hugeness, and a more direct relation between Vopěnkaness and Woodinness than already known.We also consider limits of the helices and relations with the axioms I1–I3. (shrink)

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that [Formula: see text] satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □ fails for all regular κ > ω1.

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

We explore the consistency strength of Σ31 and Σ41 absoluteness, for a variety of forcing notions.