For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of $L^1(G)$. We relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of $L^1$ functions on G.

The main scope of this short article is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. As their original axioms failed to meet natural expectations, a new axiomatization was given (i.e., Ziegler’s one), but the new axioms do not follow Messmer and Wood’s initial idea. Therefore, we aim to give a correct axiomatization that is more similar to the original one and that, as with the original axioms, involves only one Hasse–Schmidt derivation, this time based on the iterativity conditions corresponding to the Witt group.

We present an algebraic proof of the theorem stating that there are continuum many axiomatic extensions of global consequence associated with modal system E that do not admit the local deduction detachment theorem. We also prove that all these logics lack the finite frame property and have exactly three proper axiomatic extensions, each of which admits the local deduction detachment theorem.

The main question of this article is whether there is a family closed under finite differences (i.e., if A belongs to the family and $B{=}^{\ast }A$, then B also belongs to the family) that can be enumerated by any noncomputable c.e. degree, but which cannot be enumerated computably. This question was formulated by Greenberg et al. (2020) in their recent work in which families that are closed under finite differences, close to the Slaman–Wehner families, are deeply studied.

The spiritus asper as used by Frege in a letter to Russell from 1904 bears resemblance to Church’s lambda. It is natural to ask how they relate to each other. An alternative approach to functional abstraction developed by Per Martin-Löf some thirty years ago allows us to describe the relationship precisely. Frege’s spiritus asper provides a way of restructuring a unary function name in Frege’s sense such that the argument place indicator occurs all the way to the right. Martin-Löf’s alternative approach shows that this is only half of what lambda does. The other half is the deletion of the argument place indicator, resulting in what Frege would have called an isolated function name.

We consider structures of the form $(\mathbb{Z};+,C)$, where C is an additive cyclic order on $(\mathbb{Z};+)$. We show that such structures are dp-minimal and in this way produce a continuum-size family of dp-minimal expansions of $(\mathbb{Z};+)$ such that no two members of the family define the same subsets of $\mathbb{Z}$.

According to Priest, a proof can be distinct from itself in the same way that a number can. Priest does not specify any such number, so the present little note aims at filling this lacuna by providing a plain arithmetical code of a dialetheia similar to but simpler than the one presented in our recent work and thereby a natural number distinct from itself.

We show that base matrices for $\mathcal{P}(\mathit{\omega })∕\text{fin}$ of regular height larger than $\mathfrak{h}$ necessarily have maximal branches that are not cofinal. The same holds for base matrices of height $\mathfrak{h}$ if ${\mathfrak{t}}_{\mathrm{Spoiler}}<\mathfrak{h}$, where ${\mathfrak{t}}_{\mathrm{Spoiler}}$ is a variant of $\mathfrak{t}$ that has been introduced in “Construction with opposition: cardinal invariants and games” by Brendle, Hrušák, and Torres-Pérez.