These are the lecture notes from a 10-hour course that the author gave at the University of Notre Dame in September 2010. The objective of the course was to introduce some basic concepts in computable structure theory and develop the background needed to understand the author’s research on back-and-forth relations.

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are (up to provable equivalence) exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of (descriptive and Kripke) frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are (up to frame equivalence) the formulas preserved in monotonic images of (descriptive and Kripke) frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013.

The Visser rules form a basis of admissibility for the intuitionistic propositional calculus. We show how one can characterize the existence of covers in certain models by means of formulae. Through this characterization, we provide a new proof of the admissibility of a weak form of the Visser rules. Finally, we use this observation, coupled with a description of a generalization of the disjunction property, to provide a basis of admissibility for the intermediate logics ${\mathsf{BD}}_2$ and $\mathsf{GSc}$.

Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying $V=\mathsf{HOD}$. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form $\mathsf{EL}(A,P,Q)$, which asserts that $P$-definability from $A$ implies $Q$-discernibility. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set $b$ is algebraic in $a$ if it is a member of a finite set definable from $a$. Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using a similar analysis, we answer two open questions posed earlier by the third author and C. Leahy, showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

We show that the character spectrum ${Sp}_{\chi }\left(\lambda \right)$ (for a singular cardinal $\lambda$ of countable cofinality) may include any prescribed set of regular cardinals between $\lambda$ and $2^{\lambda }$.

Explicit axioms relating $◊\varphi$and $\square \varphi$ appear to be needed if $◊$ is taken to be primitive. We prove that such axioms are in fact indispensable.

We consider finite partitions of the closure $\overline{\mathcal{F}}$ of an ${\omega }^{\alpha }$-uniform barrier $\mathcal{F}$. For each partition, we get a homogeneous set having both the same combinatorial and topological structure as $\overline{\mathcal{F}}$, seen as a subspace of the Cantor space $2^{\mathbb{N}}$.

In this article, we study set mappings on 4-tuples. We continue a previous work of Komjath and Shelah by getting new finite bounds on the size of free sets in a generic extension. This is obtained by an entirely different forcing construction. Moreover, we prove a ZFC result for set mappings on 4-tuples. Also, as another application of our forcing construction, we give a consistency result for set mappings on triples.

Assume that $G$ is a definable group in a stable structure $M$. Newelski showed that the semigroup $S_G\left(M\right)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$-definable (in $M^{\mathrm{eq}}$) semigroups $S_{G,\Delta }\left(M\right)$. He also showed that it is strongly $\pi$-regular: for every $p\in S_{G,\Delta }\left(M\right)$, there exists $n\in \mathbb{N}$ such that $p^n$ is in a subgroup of $S_{G,\Delta }\left(M\right)$. We show that $S_{G,\Delta }\left(M\right)$ is in fact an intersection of definable semigroups, so $S_G\left(M\right)$ is an inverse limit of definable semigroups, and that the latter property is enjoyed by all $\infty$-definable semigroups in stable structures.

This article examines Kevin Scharp’s formal solution to the alethic paradoxes, ADT, which stands for ascending and descending truth. One of the main supposed benefits of ADT over its competitors is that it alone can validate the uses of truth concepts in theoretical contexts, such as truth-theoretic semantics. The appendixes contain a new consistency proof for ADT, and additionally show that it is conservative. As a result of its conservativity, the article argues that ADT faces a problem in accounting for certain mathematical uses of truth. Thus, Scharp’s theory needs to be amended in order to fulfill its aim of replicating all substantive uses of truth.

Gabriel Uzquiano has offered an account of indefinite extensibility for sets in the context of a modal logic. The modal operators are interpreted in terms of linguistic extensibility. After reviewing the proposal, I argue that the view should be understood as a version of in rebus structuralism about set theory. As such it is subject to the usual problems for in rebus structuralism. In particular, there is no good extra set-theoretic reason to assent to an ontology of sufficient cardinality to make true the theorems of ZFC.