Related papers: NF is Consistent
Coherence is an important aspect of text quality, and various approaches have been applied to coherence modeling. However, existing methods solely focus on a single document's coherence patterns, ignoring the underlying correlation between…
Representation stability is a theory describing a way in which a sequence of representations of different groups is related, and essentially contains a finite amount of information. Starting with Church-Ellenberg-Farb's theory of…
We lift the classical theorem of Arnol'd on homological stability for configurations spaces of the plane to the motivic world. More precisely, we prove that the schemes of unordered configurations of points in the affine line satisfy…
In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to…
Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency to describe classes of actions that respect…
Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and…
We continue the development of the theory of construction schemes over $\omega_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use…
The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on…
Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is…
The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or…
A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…
We present a novel framework for embedding hard constraint satisfaction into neural network (NN) architectures, specifically feedforward neural networks and transformers, with input-dependent affine constraints of arbitrary cardinality.…
The twisted partition monoid $\mathcal{P}_n^\Phi$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main…
In this short note, we mimic the proof of the simplicity of the theory ACFA of generic difference fields in order to provide a criterion, valid for certain theories of pure fields and fields equipped with operators, which shows that a…
Uncovering rationales behind predictions of graph neural networks (GNNs) has received increasing attention over recent years. Instance-level GNN explanation aims to discover critical input elements, like nodes or edges, that the target GNN…
This paper studies faithful explanations for Graph Neural Networks (GNNs). First, we provide a new and general method for formally characterizing the faithfulness of explanations for GNNs. It applies to existing explanation methods,…
This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that…
In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories -- whose models are sets equipped with an action by a monoid M --…
A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…
It is well known that ZFC, despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the…