Gödel’s Second Incompleteness Theorem states that no sufficiently strong consistent mathematical theory can prove its own consistency. In [1] this is proved for every axiomatic theory extending the Peano Arithmetic. For axiomatic set theory a simpler proof was given in [2] using the fact that in set theory, consistency of a set of axioms is equivalent to the existence of a model. In this note w...

During the last couple of decades, the term complexity has been commonly found in use in many fields of science, sometimes as a measurable quantity with a rigorous but narrow definition and other times as merely an ad hoc label. With an emphasis on pragmatic engineering applications, this thesis investigates the complexity concept defined in axiomatic design theory to avoid vague use of the ter...

The O$(N)$ vector model in the presence of a boundary has non-trivial fixed point $(4-\epsilon)$ dimensions and exhibits critical behaviors described by conformal field theory. spectrum operators is investigated at leading order $\epsilon$-expansion diagrammatic axiomatic approaches. In latter, we extend framework Rychkov Tan for bulk theory to case with calculate composite attention analyticit...

§ 1. The Problem of consistency in axiomatics as a logical decision problem. The state of research in the field of foundations of mathematics, to which our presentation is related, is characterized by three kinds of investigations: 1. the development of the axiomatic method, especially with the help of the foundations of geometry, 2. the founding of analysis by today’s rigorous methods through ...

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criteri...

In this invited talk, I will review five basic concepts of Axiomatic Rewriting Theory, an axiomatic and diagrammatic theory of rewriting started 25 years ago in a LICS paper with Georges Gonthier and Jean-Jacques Lévy, and developed along the subsequent years into a full-fledged 2-dimensional theory of causality and residuation in rewriting. I will give a contemporary view on the theory, inform...

About eighteen months ago I was trying to learn a little bit about model theory for a student conference (the 2003 Arizona Winter School) explaining the connections between model theory and arithmetic geometry. There are indeed deep and important connections here, and I was quickly convinced that model theory is an area of mathematics that it is good for everyone to know at least a little bit a...

We present in this paper a covariant quantization of the “massive” vector field on de Sitter (dS) space based on analyticity in the complexified pseudo-Riemanian manifold. The correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space-time is essential in our approach. We introduce the Wightman-Gärding axiomatic for vector field on ...

Executive Summary Statecharts provide for the detailed graphical description of real-time computing systems. Extended Real Time Logic (ERTL) provides for the axiomatic description of real-time computing systems. The StateMate tool provides powerful facilities for the construction and validation of Statecharts, using a conceptually simple graphical user interface. In this paper we develop a mapp...