Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [math] principles over [math]-models of [math]. They also introduced a version of this game that similarly captures provability over [math]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [math] between (...) two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [math] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [math], uncovering new differences between their logical strengths. (shrink)

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a (...) single CD theory T such that every homogeneous model of T has a PA degree. (shrink)

"By combining recent advances in the physical sciences with some of the novel ideas, techniques, and data of modern biology, this book attempts to achieve a new and different kind of evolutionary synthesis. I found it to be challenging, fascinating, infuriating, and provocative, but certainly not dull."--James H, Brown, University of New Mexico "This book is unquestionably mandatory reading not only for every living biologist but for generations of biologists to come."--Jack P. Hailman, Animal Behaviour , review of the first (...) edition "An important contribution to modern evolutionary thinking. It fortifies the place of Evolutionary Theory among the other well-established natural laws."--R.Gessink, TAXON. (shrink)

_Observing Organisations_ presents a unique approach derived from direct participant observation of small units within institutions, all in the health and social services sector. A range of contributors bring together the results of their own observational projects to show how they were able to come to a psychoanalytically informed understanding of the cultures that arise within healthcare organisations, and how this understanding can be used to overcome difficulties that arise.

Civilization; Modernism; Arab countries; intellectual life.

Contributed articles presented at Group discussion; chiefly on philosophy, Sanskrit grammar and literature.