We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators K by calculating them in the finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle $.

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial R-mecs as well as N-dimensional asymptotic classes, where N is the Morley rank of the theory.

We investigate the statement “all automorphisms of ${\mathcal {P}}(\lambda )/[\lambda ]^{<\lambda }$ are trivial.” We show that MA implies the statement for regular uncountable $\lambda <2^{\aleph _0}$, that the statement is false for measurable $\lambda $ if $2^\lambda =\lambda ^+$, and that for “densely trivial” it can be forced (together with $2^\lambda =\lambda ^{++}$) for inaccessible $\lambda $.

C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky’s construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky.

Sahlqvist theory is extended to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach’s higher-order Reverse Mathematics (RM for short), in the guise of the following principle:

$$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$

An important conceptual observation is that the usual definition of countable set—based on injections or bijections to ${\mathbb N}$—does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namely union over ${\mathbb N}$ of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are also robust, promoting the uncountability of ${\mathbb R}$ to the status of ‘big’ system in RM.

We show that under $\mathsf {BMM}$ and “there exists a Woodin cardinal,$"$ the nonstationary ideal on $\omega _1$ cannot be defined by a $\Pi _1$ formula with parameter $A \subset \omega _1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$-axiom. We further show that there are universes where $\mathsf {BPFA}$ holds and $\text {NS}_{\omega _1}$ is $\Pi _1(\{\omega _1\})$-definable. Lastly we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a generic extension of $M_1$ in which $\text {NS}_{\omega _1}$ is saturated and $\Pi _1(\{ \omega _1\} )$-definable, and $\mathsf {MA_{\omega _1}}$ holds.

Let $\mathcal {S}$ be a family of nonempty sets with VC-codensity less than $2$. We prove that, if $\mathcal {S}$ has the $(\omega ,2)$-property (for any infinitely many sets in $\mathcal {S}$, at least two among them intersect), then $\mathcal {S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal {S}$ is definable in some first-order structure, then these subfamilies can be chosen definable too.

This is a strengthening of the case $q=2$ of the definable $(p,q)$-conjecture in model theory [9] and the Alon–Kleitman–Matoušek $(p,q)$-theorem in combinatorics [6].

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP$_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q)<D_Q(p)$ for some Q, and if $q\supseteq p$ is a Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP$_1$ theories.

We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence.

Finally, we slightly modify our definitions and go beyond NSOP$_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP$_1$ and NTP$_2$ theories.

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $-jump of models of an arbitrary completion T of $\mathrm {PA}$ we show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T.

We study the $\kappa $-Borel-reducibility of isomorphism relations of complete first-order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T’, if T is classifiable and T’ is unsuperstable, then the isomorphism of models of T’ is strictly above the isomorphism of models of T with respect to $\kappa $-Borel-reducibility.

We give explicit formulas witnessing IP, IP$_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP$_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP$_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$-c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $-c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$.

The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega ,\omega }$) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty ,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.

We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $-regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $-regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold.