We characterize the linear order types $\tau $ with the property that given any countable linear order $\mathcal {L}$, $\tau \cdot \mathcal {L}$ is a computable linear order iff $\mathcal {L}$ is a computable linear order, as exactly the finite nonempty order types.

In this paper we investigate the computational complexity of deciding if the variety generated by a given finite idempotent algebra satisfies a special type of Maltsev condition that can be specified using a certain kind of finite labelled path. This class of Maltsev conditions includes several well known conditions, such as congruence permutability and having a sequence of n Jónsson terms, for some given n. We show that for such “path defined” Maltsev conditions, the decision problem is polynomial-time solvable.

We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable $\mathrm {C}^\ast $-algebras whose density character is strictly smaller than the (uncountable) cardinal invariant $\mathfrak {p}$. We show moreover that Voiculescu’s Theorem consistently fails for $\mathrm {C}^\ast $-algebras of larger density character.

We discuss the close relationship between structural theorems in (generalized) stability theory, and graph regularity theorems.

We introduce a class of notions of forcing which we call $\Sigma $-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $-Prikry. We show that given a $\Sigma $-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\Sigma $-Prikry poset that projects to $\mathbb P$ and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for $\Sigma $-Prikry posets. Putting the two works together, we obtain a proof of the following.

Theorem. If $\kappa $ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa $ remains a strong limit cardinal, every finite collection of stationary subsets of $\kappa ^+$ reflects simultaneously, and $2^\kappa =\kappa ^{++}$.

For $G$ a Polish group, we consider $G$ -flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so-called diagonal-free proofs have been given for the above-mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it.

Mauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$-null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $-ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918, he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book, he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’.

In this paper, we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal.

In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, $cl_0$, $h_0,$ and $ch_0$. We show that there exists a subset of the Baire space $\omega ^\omega ,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$-Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$. We also obtain a result on ${\mathcal {I}}$-Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum (considered on the real line ${\mathbb {R}}$) of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$, $l_0,$ and $cl_0$.

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_{2}(\mathbb{Z})$ on $\mathsf{P}^{1}(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension $d$ can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension $d+1$ .

The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\unicode[STIX]{x1D6FC}$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\unicode[STIX]{x1D6FC}$ is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When $\unicode[STIX]{x1D6FC}$ is the shift action, we characterize the class of problems for which $\unicode[STIX]{x1D6FC}$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

This article explores conditionals expressing that the antecedent makes a difference for the consequent. A ‘relevantised’ version of the Ramsey Test for conditionals is employed in the context of the classical theory of belief revision. The idea of this test is that the antecedent is relevant to the consequent in the following sense: a conditional is accepted just in case (i) the consequent is accepted if the belief state is revised by the antecedent and (ii) the consequent fails to be accepted if the belief state is revised by the antecedent’s negation. The connective thus defined violates almost all of the traditional principles of conditional logic, but it obeys an interesting logic of its own. The article also gives the logic of an alternative version, the ‘Dependent Ramsey Test,’ according to which a conditional is accepted just in case (i) the consequent is accepted if the belief state is revised by the antecedent and (ii) the consequent is rejected (e.g., its negation is accepted) if the belief state is revised by the antecedent’s negation. This conditional is closely related to David Lewis’s counterfactual analysis of causation.

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.

To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.

We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.

Hilbert 1918