Temporaryism—the view that not always everything always exists—comes in two main versions: presentism and expansionism (aka the growing block theory of time). Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterizations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterizations are full-blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete.

We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: (1) We prove that the analogue of Sacks jump inversion fails for the bounded jump and the $\mathrm{wtt}$-reducibility. (2) We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use bounded by an $\omega$-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy.

For a set $M$, let $\left|M\right|$ denote the cardinality of $M$. A family $\mathcal{F}$ is called strongly almost disjoint if there is an $n\in \omega$ such that $|A\cap B|<n$ for any two distinct elements $A$, $B$ of $\mathcal{F}$. It is shown in $\mathsf{ZF}$ (without the axiom of choice) that, for all infinite sets $M$ and all strongly almost disjoint families $\mathcal{F}\subseteq \mathcal{P}\left(M\right)$, $\left|\mathcal{F}\right|<\left|\mathcal{P}\right(M\left)\right|$ and there are no finite-to-one functions from $\mathcal{P}\left(M\right)$ into $\mathcal{F}$, where $\mathcal{P}\left(M\right)$ denotes the power set of $M$.

We prove a couple of results on NTP${}_2$ theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by $2$. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP${}_2$ theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

We analyze the descriptive complexity of several ${\Pi }_1^1$-ranks from classical analysis which are associated to Denjoy integration. We show that $\mathit{VBG}$, ${\mathit{VBG}}_{*}$, $\mathit{ACG}$, and ${\mathit{ACG}}_{*}$ are ${\Pi }_1^1$-complete, answering a question of Walsh in case of ${\mathit{ACG}}_{*}$. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most $\alpha$ steps of the transfinite process of Denjoy totalization: if $|\cdot |$ is the ${\Pi }_1^1$-rank naturally associated to $\mathit{VBG}$, ${\mathit{VBG}}_{*}$, or ${\mathit{ACG}}_{*}$, and if $\alpha <{\omega }_1^{ck}$, then $\{F\in C(I):|F|\le \alpha \}$ is ${\Sigma }_{2\alpha }^0$-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, $\left\{\right(f,F)\in M(I)\times C(I):F\in {\mathit{ACG}}_{*}\text{and}F\text{'}=f\text{a.e.}\}$ and $\{f\in M(I):f\text{is Denjoy integrable}\}$ are ${\Pi }_1^1$-complete, answering more questions of Walsh.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure $M$. We prove that if $T$ is a complete $L$-theory, then $T$ is mutually algebraic if and only if there is some model $M$ of $T$ for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory $T$ is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model $M$ of $T$.

A very natural and philosophically important subclassical logic is FDE (for first-degree entailment). This account of logical consequence can be seen as going beyond the standard two-valued account (of “just true” and “just false”) to a four-valued account (adding the additional values of “both true and false” and “neither true nor false”). A natural question arises: What account of logical consequence arises from considering further (positive) combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different (more algebraic) context. In this note we generalize Priest’s (and indirectly Shramko and Wansing’s) result to show that even if one considers ordinal-many (positive) combinations of the previous values, for any ordinal, the resulting consequence relation (the resulting logic) remains FDE.

We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.

Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched.

We show that the theory $T_{log}$ of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (have the non-independence property), this provides a new proof that $T_{log}$ is NIP.