We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $ , and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $ (where $\varphi $ is a statement in set theory), which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly $\mathsf {S4.2}$ . We also generalize this result to other related classes of forcing.

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$ , there is a model of $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ saturated and ${\Delta }_{1}$ -definable with a ladder system $\vec {C}$ and a full Suslin tree T as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article .

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the setting of InqML. We compare the expressiveness of InqML and first-order logic in the context of relational structures with two sorts, one for worlds and one for information states, and characterise inquisitive modal logic as the bisimulation invariant fragment of first-order logic over various natural classes of two-sorted structures.

We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.

We say that $\mathcal {I}$ is an ideal independent family if no element of ${\mathcal {I}}$ is a subset mod finite of a union of finitely many other elements of ${\mathcal {I}}.$ We will show that the minimum size of a maximal ideal independent family is consistently bigger than both $\mathfrak {d}$ and $\mathfrak {u},$ this answers a question of Donald Monk.

This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a variant of the Mitchell order called the internal relation.

It is known that a countable $\omega $ -categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

Henle, Mathias, and Woodin proved in [21] that, provided that ${\omega }{\rightarrow }({\omega })^{{\omega }}$ holds in a model M of ZF, then forcing with $([{\omega }]^{{\omega }},{\subseteq }^*)$ over M adds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model $M[\mathcal {U}]$ , where $\mathcal {U}$ is a Ramsey ultrafilter, with many properties of the original model M. This begged the question of how important the Ramseyness of $\mathcal {U}$ is for these results. In this paper, we show that several classes of $\sigma $ -closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme, k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras $\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$ , $2\le {\alpha }<{\omega }_1$ , forcing non-p-points also produce barren extensions.

A wide Aronszajn tree is a tree of size and height $\omega _{1}$ with no uncountable branches. We prove that under $MA(\omega _{1}\!)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994.

We also prove that under $MA(\omega _{1}\!)$ , every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under $MA(\omega _{1}\!)$ every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under $MA(\omega _{1}\!)$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under $MA(\omega _{1}\!)$ there are no universal Aronszajn trees.

We characterize Weihrauch reducibility in $ \operatorname {\mathrm {E-PA^{\omega }}} + \operatorname {\mathrm {QF-AC^{0,0}}}$ and all systems containing it by the provability in a linear variant of the same calculus using modifications of Gödel’s Dialectica interpretation that incorporate ideas from linear logic, nonstandard arithmetic, higher-order computability, and phase semantics.

We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$ , and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $ , then it is not the case that ${\mathord {\mathbb P}}$ forces that T has such a branch. $\Sigma ^1_1$ -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega $ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$ .

We prove that if an $\omega $ -categorical structure has an $\omega $ -categorical homogeneous Ramsey expansion, then so does its model-complete core.

We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal {M}$ , or an axiomatization S thereof, we find a modal logic M such that a modal sentence $\varphi $ is a theorem of M if and only if the sentence $\varphi ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal {M}$ or a theorem of S under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of noncongruent modal logics whose internal logic is nonclassical with respect to this semantics.

It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes. The extent to which resurrection and maximality overlap is explored via the local maximality principle.

For every $n\in \omega \setminus \{0,1\}$ we introduce the following weak choice principle:

Moreover, we consider the following choice principle:

We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace $\mathcal {F}$ by a linearly ordered or a well-ordered family.