Related papers: Normalization of IZF with Replacement
The standard treatment of sets and definable classes in first-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between objects and concepts. Nevertheless, in set…
In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms…
Isabelle is a generic theorem prover, designed for interactive reasoning in a variety of formal theories. At present it provides useful proof procedures for Constructive Type Theory, various first-order logics, Zermelo-Fraenkel set theory,…
In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified…
Set theory is widely believed to provide a secure foundation for deductive mathematics, but current set theories do not quite do this. The mainstream essentially uses na\"\i ve set theory. After Russell's paradox showed this to be…
Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a…
We present intersection type systems in the style of sequent calculus, modifying the systems that Valentini introduced to prove normalisation properties without using the reducibility method. Our systems are more natural than Valentini's…
CZF is a system of set theory which, over classical logic, is equivalent to ZF, while over intuitionistic logic, it has a well-known constructive type-theoretic interpretation. This article introduces a simpler, intuitive family of…
Constructive theories usually have interesting metamathematical properties where explicit witnesses can be extracted from proofs of existential sentences. For relational theories, probably the most natural of these is the existence…
We introduce a simple natural deduction system for reasoning with judgments of the form "there exists a proof of $\varphi$" to explore the notion of judgmental existence following Martin-L\"{o}f's methodology of distinguishing between…
We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on…
This thesis develops a framework for formalizing reasoning about specifications of systems written in LF. This formalization centers around the development of a reasoning logic that can express the sorts of properties which arise in…
In this article, we study the multiple zeta functions (MZF) and some of its variants at identical arguments. Using the harmonic product, these functions can be expressed as polynomials in the Riemann zeta function. Firstly, we note that an…
We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from…
We use the technique of "classical realizability" to build new models of ZF + DC in which R is not well ordered. This gives new relative consistency results, probably not obtainable by forcing. This gives also a new method to get programs…
For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…
In this paper, we provide a positive answer to the question of Matthews whether $\mathsf{ZF}^-$ is consistent with a non-trivial cofinal Reinhardt elementary embedding $j\colon V\to V$. The consistency follows from $\mathsf{ZFC} + I_0$, and…
We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus…
The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic. We will see that Heyting arithmetic is bi-interpretable with $\mathsf{CZF^{fin}}$, the finitary version…
This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $\lambda$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using…