If X0 and X1 are both generic, the theories of the degrees below X0 and X1 are the same. The same is true if both are random. We show that the n-genericity or n-randomness of X do not suffice to guarantee that the degrees below X have these common theories. We also show that these two theories (for generics and randoms) are different. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis.

We characterize, in terms of determinacy, the existence of the least inner model of “every object of type k has a sharp.” For k ∈ ω, we define two classes of sets, and , which lie strictly between and . Let #k be the (partial) sharp function on objects of type k. We show that the determinacy of follows from

L[#k] ⊨ “every object of type k has a sharp”,

and we show that the existence of indiscemibles for L[#k] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of .

Assuming 0# does not exist, we present a combinatorial approach to Jensen's method of coding by a real. The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

Vardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$—the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$.

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

I continue the study of the blurry HOD hierarchy. The technically most involved result is that the theory ZFC + “$\aleph _\omega $ is a strong limit cardinal and $\aleph _{\omega +1}$ is the least leap” is equiconsistent with the theory ZFC + “there is a measurable cardinal.”

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $-nonsplitting. The following holds:

1. 1. When $\mu $-nonforking is restricted to $(\mu ,\geq \chi )$-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This generalizes Vasey’s result [37, Corollary 13.16] which assumed $\mu $-superstability to obtain same properties but with local character $\aleph _0$.

2. 2. There is $\lambda \in [\mu ,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $-symmetry while $\mu $-nonforking in (1) has symmetry. In this case:

   1. (a) $\mathbf {K}$ has the uniqueness of $(\mu ,\geq \chi )$-limit models: if $M_1,M_2$ are both $(\mu ,\geq \chi )$-limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$;

   2. (b) any increasing chain of $\mu ^+$-saturated models of length $\geq \chi $ has a $\mu ^+$-saturated union. These generalize [31] and remove the symmetry assumption in [10, 38] .

Under $(<\mu )$-tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $-local character of $\mu $-nonsplitting.

Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank.

In this article we define a logical system called Hybrid Partial Type Theory ($\mathcal {HPTT}$). The system is obtained by combining William Farmer’s partial type theory with a strong form of hybrid logic. William Farmer’s system is a version of Church’s theory of types which allows terms to be non-denoting; hybrid logic is a version of modal logic in which it is possible to name worlds and evaluate expressions with respect to particular worlds. We motivate this combination of ideas in the introduction, and devote the rest of the article to defining, axiomatising, and proving a completeness result for $\mathcal {HPTT}$.