This technical note shows how we have combined prescriptive type checking and constraint solving to increase automation during software verification. We do so by defining a type system and implementing a typechecker for $\{log\}$ (read ‘setlog’), a Constraint Logic Programming language and satisfiability solver based on set theory. The constraint solver is proved to be safe w.r.t. the type system. Two industrial-strength case studies are presented where this combination is used with very good results.

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands).

By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with a ZF-like theory.

This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974).

On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which $\Gamma _0$ and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of $\Gamma _0$ as the “limit of predicativity”.

This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how those equivalence relations are themselves related to each other. We show that Morita equivalence fits into this framework and provide answers to questions left open in Barrett and Halvorson [4]. We conclude by setting up a diagram of known relationships and leave open some questions for future work.

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness.

Tangled Type Theory was introduced by Randall Holmes in [3] as a new way of approaching the consistency problem for $\mathrm {NF}$. Although the task of finding models for this theory is far from trivial (considering it is equiconsistent with $\mathrm {NF}$), ways of constructing models for certain fragments of it have been discovered. In this article, we present a simpler way of constructing models of predicative Tangled Type Theory and consequently of predicative $\mathrm {NF}$. In these new models of predicative $\mathrm {NF}$, the universe is well-orderable and equinumerous to the set of singletons.

In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$.

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.

We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple $(U,\mathcal S,\mathcal C)$, where U is a non-empty set (“the universe”), $\mathcal S$ is a perfect Hausdorff topology on U, and $\mathcal C\subseteq \mathcal P(U)$ together with $\mathcal S$ satisfy certain axioms.

We refute a conjecture put forward by Hoffmann-Jørgensen, who conjectured that the existence of a “complete” Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology.

On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe.

Further, we investigate two new cardinal invariants $\mathfrak u_{M}$ and $\mathfrak u_{T}$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show $\mathfrak u_{M}=\mathfrak u_{T}=2^{\aleph _{0}}$ follows from Martin’s Axiom, and, contrastingly, we show that $\aleph _{1}=\mathfrak u_{M}=\mathfrak u_{T}<2^{\aleph _{0}}=\aleph _{2}$ in the Baumgartner–Laver model.

Finally, consequences for psychology are discussed.

Formal reasoning about finite sets and cardinality is important for many applications, including software verification, where very often one needs to reason about the size of a given data structure. The Constraint Logic Programming tool $$\{ log\} $$ provides a decision procedure for deciding the satisfiability of formulas involving very general forms of finite sets, although it does not provide cardinality constraints. In this paper we adapt and integrate a decision procedure for a theory of finite sets with cardinality into $$\{ log\} $$ . The proposed solver is proved to be a decision procedure for its formulas. Besides, the new CLP instance is implemented as part of the $$\{ log\} $$ tool. In turn, the implementation uses Howe and King’s Prolog SAT solver and Prolog’s CLP(Q) library, as an integer linear programming solver. The empirical evaluation of this implementation based on +250 real verification conditions shows that it can be useful in practice.

Under consideration in Theory and Practice of Logic Programming (TPLP)