相关论文: From Incompleteness Towards Completeness

It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…

We formally prove the existence of an enduring incongruence pervading a widespread interpretation of the Bell inequality and explain how to rationally avoid it with a natural assumption justified by explicit reference to a mathematical…

G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum…

It has been commonly argued, on the basis of Goedel's theorem and related mathematical results, that true artificial intelligence cannot exist. Penrose has further deduced from the existence of human intelligence that fundamental changes in…

A notion of delegated causality is introduced. This subtle kind of causality is dual to interventional causality. Delegated causality elucidates the causal role of dynamical systems at the "edge of chaos", explicates evident cases of…

General relativity, despite its profound successes, fails as a complete theory due to presence of singularities. While it is widely believed that quantum gravity has the potential to be a complete theory, in which spacetime consistently…

These notes were written for a presentation given at the university Paris VII in January 2012. The goal was to explain a proof of a famous theorem by P. Deligne about coherent topoi (coherent topoi have enough points) and to show how this…

The set-theoretical model of Goedel's system T is not fully abstract. We also briefly discuss fully abstract models of system T.

We initiate an investigation how the fundamental concept of independence can be represented effectively in the presence of incomplete information in relational databases. The concepts of possible and certain independence are proposed, and…

The article proposes a new technique for proving the undefinability of logical connectives through each other and illustrates the technique with several examples. Some of the obtained results are new proofs of the existing theorems, others…

The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry.

This is a paper for a special issue of the journal "Studia Semiotyczne" devoted to Stanislaw Krajewski's paper [30]. This paper gives some supplementary notes to Krajewski's [30] on the Anti-Mechanist Arguments based on G\"{o}del's…

Standard interpretations of Goedel's "undecidable" proposition, [(Ax)R(x)], argue that, although [~(Ax)R(x)] is PA-provable if [(Ax)R(x)] is PA-provable, we may not conclude from this that [~(Ax)R(x)] is PA-provable. We show that such…

The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…

Irreversible phenomena, such as the production of entropy and heat, arise from fundamental reversible dynamics because the forward dynamics is too complex, in the sense that it becomes impossible to provide the necessary information to keep…

We take an argument of G\"odel's from his ground-breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: the sentence $G$ says about itself that it is not provable, and $G$ is indeed not provable;…

Emergence is a pregnant property in various fields. It is the fact for a phenomenon to appear surprisingly and to be such that it seems at first sight that it is not possible to predict its apparition. That is the reason why it has often…

In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process…

We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.

Goedel's explicit thesis was that his undecidable formula GUS is a well-formed, well-defined formal sentence in any formalisation of Intuitive Arithmetic IA in which the axioms and rules of inference are recursively definable. His implicit…