We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...) purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

John Wilson's work as moral educator is summarized and evaluated. His rationalist humanistic approach is based on a componential characterization of the morally educated person. Such a person consistently manifests a unity of reflection, feeling, belief, and acting under the logically structured rubrics of PHIL, EMP, GIG and KRAT, and exemplifying the formal features of 'moral opinion'. The rationale and conceptual status of the components is discussed, as is the view that the concept of education entails that teachers be moral (...) educators. This involves cultivating autonomous rationality with respect to the unconscious, motivation, day-to-day moral decision-making, and the emotions; in the latter case there are extensive applications in religious education. Finally, certain weaknesses and pre-eminent strengths of Wilson's position are indicated, and comparisons briefly made with the views of McPhail, Peters, Frankena and Kohlberg. (shrink)

Principia Mathematica was the first sustained and successful formalization of mathematics.

Following an introduction to noise and noise regulation of wind turbines, the problem of adverse health effects of turbine noise is discussed. This is attributed to the characteristics of turbine noise and deficiencies in the regulation of this noise. Both onshore and offshore wind farms are discussed.