Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions.

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $-formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

We investigate Steel’s conjecture in ‘The Core Model IterabilityProblem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$-iterable, $1$-small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that thereis an iterable premouse M which models KP and which has a-Woodin cardinal. On the other hand, we show thatassuming there is no transitive model of KP with a Woodin cardinal theconjecture holds. In the course of this we will also show that ifM is a premouse which models KP with a largest, regular,uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibilityunder iteration maps. At last, we will prove a fact about the closure of the setof ordinals at which a weasel has the S-hull property. Thisanswers another question implicit in remarks in [10].

We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of

$$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

We study the possible structures which can be carried by sets which have no countable subset, but which fail to be ‘surjectively Dedekind finite’, in two possible senses, that there is surjection to $\omega $, or alternatively, that there is a surjection to a proper superset.

We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $ \aleph _1 $-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the set $D'$ comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation $D \mapsto D'\ n$ times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden $2$, we show that any definable unary discrete set must be definable in some elementary extension of the structure $\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).

We continue investigating variants of the splitting and reaping numbers introduced in [4]. In particular, answering a question raised there, we prove the consistency of and of . Moreover, we discuss their natural generalisations $\mathfrak {s}_{\rho }$ and $\mathfrak {r}_{\rho }$ for $\rho \in (0,1)$, and show that $\mathfrak {r}_{\rho }$ does not depend on $\rho $.

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$.

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.