Related papers: Set-theoretic mereology
We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of…
A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much…
The aim of this paper is to give mathematical account of an argument of David Lewis in Parts of Classes in defense of universalism in mereology. Specifically we study how to extend models of Core Mereology (following Achille Varzi's…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
We see how nested sequents, a natural generalisation of hypersequents, allow us to develop a systematic proof theory for modal logics. As opposed to other prominent formalisms, such as the display calculus and labelled sequents, nested…
In this paper, we argue that while the concept of a set-theoretic paradox (or paradoxical set) can be relatively well-defined within a formal setting, the concept of a set-theoretic hypodox (or hypodoxical set) remains significantly less…
We investigate the position that foundational theories should be modelled on ordinary computability. In this context, we investigate the metamathematics of $\Sigma$ formulas. We consider theories whose axioms are implications between…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using…
In earlier work, we introduced flexible inference and decision-theoretic metareasoning to address the intractability of normative inference. Here, rather than pursuing the task of computing beliefs and actions with decision models composed…
In this research a new algebraic semantics of rough set theory including additional meta aspects is proposed. The semantics is based on enhancing the standard rough set theory with notions of 'relative ability of subsets of approximation…
Stanis{\l}aw Le\'sniewski's mereology was formulated in a specific way, deviating from standard formalizations. Nowadays, Le\'sniewski's theory is presented in the form of an elementary theory or translated into the language of the theory…
We consider the problem of formalizing the familiar notion of widening in abstract interpretation in higher-order logic. It turns out that many axioms of widening (e.g. widening sequences are ascending) are not useful for proving…
We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to…
A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists,…
While model checking has often been considered as a practical alternative to building formal proofs, we argue here that the theory of sequent calculus proofs can be used to provide an appealing foundation for model checking. Since the…
In this paper, we propose a set theoretic approach for knowledge representation. While the syntax of an application domain is captured by set theoretic constructs including individuals, concepts and operators, knowledge is formalized by…
The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order…