We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $-bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where T is an $\omega _1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$, and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $, and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $. A non-normal modal logic $\mathbf {D}$, which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$. A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$, and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $. Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$, then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$.

The class of all $\ast $-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras—ranging from the equational theory to the Horn one, with restricted fragments of the latter in between—was analyzed by Kozen (2002). This paper deals with similar problems for $\ast $-continuous residuated Kleene lattices, also called $\ast $-continuous action lattices, where the product operation is augmented by residuals. We prove that, in the presence of residuals, the fragment of the corresponding Horn theory with $\ast $-free hypotheses has the same complexity as the $\omega ^\omega $ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions are allowed as hypotheses, then the complexity drops down to $\Pi ^0_1$ (i.e., the complement of the halting problem), which is the same as that for $\ast $-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and our upper bounds are obtained for the latter ones.

This paper provides a consistent first-order theory solving the knower paradoxes of Kaplan and Montague, with the following main features: 1. It solves the knower paradoxes by providing a faithful formalization of the principle of veracity (that knowledge requires truth), using both a knowledge and a truth predicate. 2. It is genuinely untyped i.e., it is untyped not only in the sense that it uses a single knowledge predicate applying to all sentences in the language (including sentences in which this predicate occurs), but in the sense that its axioms quantify over all sentences in the language, thus supporting comprehensive reasoning with untyped knowledge ascriptions. 3. Common knowledge predicates can be defined in the system using self-reference. These facts, together with a technique based on Löb’s theorem, enables it to support comprehensive reasoning with untyped common knowledge ascriptions (without having any axiom directly addressing common knowledge).

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

We study versions of the tree pigeonhole principle, $\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether $\mathsf {TT}^1$ is $\Pi ^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike $\mathsf {RT}^1$, the problem $\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf {TT}^1$.

We prove a full measurable version of Vizing’s theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal {G}$ of degree uniformly bounded by $\Delta \in \mathbb {N}$ defined on a standard probability space $(X,\mu )$ admits a $\mu $-measurable proper edge coloring with $(\Delta +1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)], who derived the same conclusion under the additional assumption that the measure $\mu $ is $\mathcal {G}$-invariant.

We explore general notions of consistency. These notions are sentences $\mathcal {C}_{\alpha }$ (they depend on numerations $\alpha $ of a certain theory) that generalize the usual features of consistency statements. The following forms of consistency fit the definition of general notions of consistency (${\texttt {Pr}}_{\alpha }$ denotes the provability predicate for the numeration $\alpha $): $\neg {\texttt {Pr}}_{\alpha }(\ulcorner \perp \urcorner )$, $\omega \text {-}{\texttt {Con}}_{\alpha }$ (the formalized $\omega $-consistency), $\neg {\texttt {Pr}}_{\alpha }(\ulcorner {\texttt {Pr}}_{\alpha }(\ulcorner \cdots {\texttt {Pr}}_{\alpha }(\ulcorner \perp \urcorner )\cdots \urcorner )\urcorner )$, and $n\text {-}{\texttt {Con}}_{\alpha }$ (the formalized n-consistency of Kreisel).

We generalize the former notions of consistency while maintaining two important features, to wit: Gödel’s Second Incompleteness Theorem, i.e., (with $\xi $ some standard $\Delta _0(T)$-numeration of the axioms of T), and a result by Feferman that guarantees the existence of a numeration $\tau $ such that $T\vdash \mathcal {C}_\tau $.

We encompass slow consistency into our framework. To show how transversal and natural our approach is, we create a notion of provability from a given $\mathcal {C}_{\alpha }$, we call it $\mathcal {P}_{\mathcal {C}_{\alpha }}$, and we present sufficient conditions on $\mathcal {C}_{\alpha }$ for the notion $\mathcal {P}_{\mathcal {C}_{\alpha }}$ to satisfy the standard derivability conditions. Moreover, we also develop a notion of interpretability from a given $\mathcal {C}_{\alpha }$, we call it $\rhd _{\mathcal {C}_{\alpha }}$, and we study some of its properties. All these new notions—of provability and interpretability—serve primarily to emphasize the naturalness of our notions, not necessarily to give insights on these topics.

A central topic in mathematical logic is the classification of theorems from mathematics in hierarchies according to their logical strength. Ideally, the place of a theorem in a hierarchy does not depend on the representation (aka coding) used. In this paper, we show that the standard representation of compact metric spaces in second-order arithmetic has a profound effect. To this end, we study basic theorems for such spaces like a continuous function has a supremum and a countable set has measure zero. We show that these and similar third-order statements imply at least Feferman’s highly non-constructive projection principle, and even full second-order arithmetic or countable choice in some cases. When formulated with representations (aka codes), the associated second-order theorems are provable in rather weak fragments of second-order arithmetic. Thus, we arrive at the slogan that

$$ \begin{align*} {coding\ compact\ metric\ spaces\ in\ the\ language\ of\ second\text{-}order\ arithmetic\ can\ be\ as\ hard }\\ {as\ second\text{-}order\ arithmetic\ or\ countable\ choice}. \end{align*} $$

We believe every mathematician should be aware of this slogan, as central foundational topics in mathematics make use of the standard second-order representation of compact metric spaces. In the process of collecting evidence for the above slogan, we establish a number of equivalences involving Feferman’s projection principle and countable choice. We also study generalisations to fourth-order arithmetic and beyond with similar-but-stronger results.

We investigate natural variations of behaviourally correct learning and explanatory learning—two learning paradigms studied in algorithmic learning theory—that allow us to “learn” equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\boldsymbol {\Pi }^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. We first established two results:

1. (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$.

2. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b)$E(G)\leq _B E_0^\omega $; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean.

A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le _BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker (S)$ is non-archimedean, where $G_0$ is the connected component of the identity of G. The converse holds if G is connected, $S(G)$ is closed in H, and the interval $[0,1]$ can be embedded into H.

As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively.

The aim of this paper is to give a full exposition of Leibniz’s mereological system. My starting point will be his papers on Real Addition, and the distinction between the containment and the part-whole relation. In the first part (§2), I expound the Real Addition calculus; in the second part (§3), I introduce the mereological calculus by restricting the containment relation via the notion of homogeneity which results in the parthood relation (this corresponds to an extension of the Real Addition calculus via what I call the Homogeneity axiom). I analyze in detail such a notion, and argue that it implies a gunk conception of (proper) part. Finally, in the third part (§4), I scrutinize some of the applications of the containment-parthood distinction showing that a number of famous Leibnizian doctrines depend on it.

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set of objects whatsoever. In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation.

We analyze a countable support product of a free Suslin tree which turns it into a highly rigid Kurepa tree with no Aronszajn subtree. In the process, we introduce a new rigidity property for trees, which says roughly speaking that any non-trivial strictly increasing function from a section of the tree into itself maps into a cofinal branch.

We show that in the Silver model the inequality $\mathrm {cov}(\mathfrak {C} _2) < \mathrm {cov}(\mathfrak {P}_2)$ holds true, where $\mathfrak {C}_2$ and $\mathfrak {P}_2$ are the two-dimensional Mycielski ideals.

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065–1092]. For an ideal $\mathcal {I}$ on $\omega $ we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ and write $f\leq _{\mathcal {I}} g$ if $\{n\in \omega :f(n)>g(n)\}\in \mathcal {I}$, where $f,g\in \omega ^{\omega }$.

We study the cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ describing the smallest sizes of subsets of $\mathcal {D}_{\mathcal {I}}$ that are unbounded from below with respect to $\leq _{\mathcal {I}}$.

In particular, we examine the relationships of $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ with the dominating number $\mathfrak {d}$. We show that, consistently, $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))>\mathfrak {d}$ for some ideal $\mathcal {I}$, however $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))\leq \mathfrak {d}$ for all analytic ideals $\mathcal {I}$. Moreover, we give example of a Borel ideal with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\operatorname {\mathrm {add}}(\mathcal {M})$.

A left-variable word over an alphabet A is a word over $A \cup \{\star \}$ whose first letter is the distinguished symbol $\star $ standing for a placeholder. The ordered variable word theorem ($\mathsf {OVW}$), also known as Carlson–Simpson’s theorem, is a tree partition theorem, stating that for every finite alphabet A and every finite coloring of the words over A, there exists a word $c_0$ and an infinite sequence of left-variable words $w_1, w_2, \dots $ such that $\{ c_0 \cdot w_1[a_1] \cdot \dots \cdot w_k[a_k] : k \in \mathbb {N}, a_1, \dots , a_k \in A \}$ is monochromatic.

In this article, we prove that $\mathsf {OVW}$ is $\Pi ^0_4$-conservative over $\mathsf {RCA}_0 + \mathsf {B}\Sigma ^0_2$. This implies in particular that $\mathsf {OVW}$ does not imply $\mathsf {ACA}_0$ over $\mathsf {RCA}_0$. This is the first principle for which the only known separation from $\mathsf {ACA}_0$ involves non-standard models.

We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which may be of independent interest.