Related papers: Proof Theory
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…
"Ever since the advent of modern quantum mechanics in the late 1920's, the idea has been prevalent that the classical laws of probability cease, in some sense, to be valid in the new theory. [...] The primary object of this presentation is…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
Proof Designer is a computer software program designed to help Mathematics students learn to write mathematical proofs. Under the guidance of the user, Proof Designer assists in writing outlines of proofs in elementary set theory. Proof…
This paper revisits the foundations of mathematical proof through the lens of Aristotle's threefold conception of truth: sensory evidence, axiomatic definition, and syllogistic deduction. I argue that modern mathematics has too often…
This talk is a sneak preview of the project, 'proof theory for theories of ordinals'. Background, aims, survey and furture works on the project are given. Subsystems of second order arithmetic are embedded in recursively large ordinals and…
Interactive proof assistants make it possible for ordinary mathematicians to write definitions and theorems in a formal proof language, like a programming language, so that a computer can parse them and check them against the rules of a…
A step-by-step presentation of the code for a small theorem prover introduces theorem-proving techniques. The programming language used is Standard ML. The prover operates on a sequent calculus formulation of first-order logic, which is…
Prawitz conjectured that the proof-theoretically valid logic is intuitionistic logic. Recent work on proof-theoretic validity has disproven this. In fact, it has been shown that proof-theoretic validity is not even closed under…
Proof theory provides a foundation for studying and reasoning about programming languages, most directly based on the well-known Curry-Howard isomorphism between intuitionistic logic and the typed lambda-calculus. More recently, a…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
We report on several scenarios of using automated theorem proving software in university education. In particular, we focus on using the Theorema system in a software-enhanced logic-course for students in computer science or artificial…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
D'Alembert made the first serious attempt to prove the Fundamental Theorem of Algebra (FTA) in 1746. An elementary proof of (FTA) based on the same idea is given in Proofs from THE BOOK. We give a shorter and more transperant version of…
In an emerging computing paradigm, computational capabilities, from processing power to storage capacities, are offered to users over communication networks as a cloud-based service. There, demanding computations are outsourced in order to…
Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative…
This paper presents Holbert: a work-in-progress pedagogical proof assistant and online textbook platform, aimed at the educational use-case, specifically for the teaching of programming language theory. Holbert allows proof exercises and…
Argumentation theory is a powerful paradigm that formalizes a type of commonsense reasoning that aims to simulate the human ability to resolve a specific problem in an intelligent manner. A classical argumentation process takes into account…