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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)

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

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)

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.