Related papers: Set Theory with Urelements

This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even…

Ranking entities such as algorithms, devices, methods, or models based on their performances, while accounting for application-specific preferences, is a challenge. To address this challenge, we establish the foundations of a universal…

The eigenspace theory of unitary reflection groups, initiated by Springer and Lehrer, suggests that the following object is worthy of study: the poset of eigenspaces of elements of a unitary reflection group, for a fixed eigenvalue, ordered…

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself),…

In "Object generators, relaxed sets, and a foundation for mathematics", we introduced ``object generators'', a logical environment much more general than set theory. Inside this we found a `relaxed' version of set theory. That paper is…

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,…

In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The…

In this paper we develop a theory called $\sigma$-Set Theory, in which we present an axiom system developed from the study of Set Theories of Zermelo-Fraenkel, Neumann-Bernays-Godel and Morse-Kelley. In $\sigma$-Set Theory, we present the…

We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection…

This paper presents new results from a detailed study of the structure of autocatalytic sets. We show how autocatalytic sets can be decomposed into smaller autocatalytic subsets, and how these subsets can be identified and classified. We…

The study is aimed at revealing the most important substructures (fragments) of polyenes with heteroatoms determining the alteration in the conjugation energy of the whole compound due to substitution and the relevant charge redistribution.…

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field,…

The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is…

This article presents the most interesting philosophical issues as they arise in causal set theory. The first concerns the apparent disappearance of spacetime at the fundamental level. It shows how the looming empirical incoherence is…

Flexoelectricity is characterised by the coupling of the gradient of the deformation and the electrical polarization in a dielectric material. A novel micromorphic approach is presented to accommodate the resulting higher-order gradient…

The goal of confidence-set learning in the binary classification setting is to construct two sets, each with a specific probability guarantee to cover a class. An observation outside the overlap of the two sets is deemed to be from one of…

We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large…

We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a…

Fairly deep results of Zermelo-Frenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K = K,…

We present a system for bottom-up cumulative learning of myriad concepts corresponding to meaningful character strings, and their part-related and prediction edges. The learning is self-supervised in that the concepts discovered are used as…