Gödel's incompleteness theorems

Summary

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. Employing a diagonal argument, Gödel's incompletenes

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https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems

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In this thesis we will treat the Dirichlet problem for systems of implicit equations, i.e. where Ω ⊂ Rn is an open set, u : Ω → Rm, Fi : Ω × Rm × Rm×n → R, m,n ≥ 1, are continuous functions and φ, the boundary datum, is given. At first we will be interested in the study of problems of the type (1) under constraints. We will show theorems of existence of Lipschitz solutions using an approach based on Baire category theorem. As corollaries of our abstract results, we will give two theorems related to the constraints det Du > 0 and det Du = 1, the constraints most closely related to the applications. Indeed, the constraints det Du > 0 and det Du = 1 come from nonlinear elasticity and represent respectively the conditions of non interpenetration of matter and incompressibility. In the second part of this study, we will focus on the applications. We will treat many examples such as the case of singular values, potential wells under the incompressibility constraint (i.e. det Du = l), the problem of confocal ellipses, the problem of nematic elastomers, particularly related to the fields of nonlinear elasticity, the microstructure of the crystals and the optimal design, as well as the complex eikonal equation (application related to geometric optics). From the mathematical point of view, we will give sufficient conditions of solvability of the system (1). These conditions consist in characterizing the different convex hulls. Indeed, the possibility of representing these sets, in algebraic terms, gives one of the conditions which the boundary datum must satisfy so that a problem of the type (1) admits a solution. Finally we will extend to polyconvex sets properties such as the gauge, the characterization of the extreme points and the Choquet function, all of which are well-known tools within the framework of classical convex analysis.

EPFL2000

The special linear group G = SLn(Z[x(1), ... , x(k)]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, and p be any real number in (1, infinity). The main result is the following: any finite index subgroup of G has the fixed point property with respect to every affine isometric action on the space of p-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are a generalization of previous theorems respectively of the author and of Bader-Furman-Gelander-Monod, which treated a commutative L-p-setting.

Amer Mathematical Soc2013

Node Attribute Completion in Knowledge Graphs with Multi-Relational Propagation

Eda Bayram, Alberto Garcia Duran, Robert West

The existing literature on knowledge graph completion mostly focuses on the link prediction task. However, knowledge graphs have an additional incompleteness problem: their nodes possess numerical attributes, whose values are often missing. Our approach, denoted as MrAP, imputes the values of missing attributes by propagating information across the multi-relational structure of a knowledge graph. It employs regression functions for predicting one node attribute from another depending on the relationship between the nodes and the type of the attributes. The propagation mechanism operates iteratively in a message passing scheme that collects the predictions at every iteration and updates the value of the node attributes. Experiments over two benchmark datasets show the effectiveness of our approach.

2020