We consider implicit definability over the natural number system $\mathbb{N},+,\times ,=$. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of $\mathbb{N}$ which are not explicitly definable from each other. The second theorem says that there exists a subset of $\mathbb{N}$ which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of $\mathbb{N}$. Previous proofs of these theorems have used finite- or infinite-injury priority constructions. Our new proof is easier in that it uses only a nonpriority oracle construction, adapted from the standard proof of the Friedberg jump theorem.

The article studies properties of functional dependencies between strategies of players in Nash equilibria of multiplayer strategic games. The main focus is on the properties of functional dependencies in the context of a fixed dependency graph for payoff functions. A logical system describing properties of functional dependence for any given graph is proposed and is proven to be complete.

The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey.

Specifically, we prove that (i) every algebraic real is spectral, (ii) every automatic real is spectral, (iii) the subword density of a spectral real is either 0 or 1, and both may occur, and (iv) every right-computable real number between 0 and 1 occurs as the subword entropy of a spectral real.

In addition, Durand et al. note that the set of spectral reals is not closed under addition or multiplication. We extend this result by showing that the class of spectral reals is not closed under any computable operation satisfying some mild conditions.

We give a direct and elementary proof of the fact that every real-valued probability measure can be approximated—up to an infinitesimal—by a hyperreal-valued one which is regular and defined on the whole powerset of the sample space.

We study properties of certain subclasses of the Dedekind finite sets (addressed as finiteness classes) in set theory without the axiom of choice (AC) with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows:

1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of its elements are comparable.

2. The principle “Small Violations of Choice” (SVC)—introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above.

3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF.

4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above.

5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above.

A computable structure $\mathcal{A}$ is $\mathbf{x}$-computably categorical for some Turing degree $\mathbf{x}$ if for every computable structure $\mathcal{B}\cong \mathcal{A}$ there is an isomorphism $f:\mathcal{B}\to \mathcal{A}$ with $f{\le }_T\mathbf{x}$. A degree $\mathbf{x}$ is a degree of categoricity if there is a computable structure $\mathcal{A}$ such that $\mathcal{A}$ is $\mathbf{x}$-computably categorical, and for all $\mathbf{y}$, if $\mathcal{A}$ is $\mathbf{y}$-computably categorical, then $\mathbf{x}{\le }_T\mathbf{y}$.

We construct a ${\Sigma }_2^0$ set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.

Continuous extension (CE) cell decomposition in o-minimal structures was introduced by Simon Andrews to establish the open cell property (OCP) in those structures. Here, we define strong $\mathrm{CE}$-cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has $\mathrm{SCE}$-cell decomposition if and only if its canonical o-minimal extension has $\mathrm{CE}$-cell decomposition. Then, we show that every weakly o-minimal structure with $\mathrm{SCE}$-cell decomposition satisfies $\mathrm{OCP}$. Our last result implies that every o-minimal structure in which every definable open set is a union of finitely many open $\mathrm{CE}$-cells, has $\mathrm{CE}$-cell decomposition.

In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity—by the splitters and by the adjointness—can be explicated with the help of the axiom of choice as well.

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every (pointwise) algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only the sets that are definable over what has been built so far, but also those that are algebraic (or, equivalently, implicitly definable) over the existing structure. While we know that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open.