By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$ . We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has (...) a certain "slowness" property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ∈ ε there exists B in the orbit of A such that X ≤ T B under relative Turing computability (≤ T ). We produce B using the Δ 0 3 -automorphism method we introduced earlier. (shrink)

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that (...) X is $low_2$ if $X\prime\prime$ $\leq_{T} 0\prime$ . To prove that a $low_2$ set X is not prime bounding we use a $0\prime$ -computable listing of the array of sets { Y : Y $\leq_{T}$ X } to build a CAD theory T which diagonalizes against all potential X-decidable prime models U of T. To prove that any $non-low_{2}$ ; X is indeed prime bounding, we fix a function f $\leq_T$ X that is not dominated by a certain $0\prime$ -computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula $\varphi (\bar{x})$ consistent with T, a principal type which contains it, and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory. (shrink)

It is shown that for every nonzero r.e. degree c there is a linear ordering of degree c which is not isomorphic to any recursive linear ordering. It follows that there is a linear ordering of low degree which is not isomorphic to any recursive linear ordering. It is shown further that there is a linear ordering L such that L is not isomorphic to any recursive linear ordering, and L together with its ‘infinitely far apart’ relation is of low (...) degree. Finally, an analogue of the recursion theorem for recursive linear orderings is refuted. (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)

Music can reduce stress and anxiety, enhance positive mood, and facilitate social bonding. However, little is known about the role of music and related personal or cultural variables in maintaining wellbeing during times of stress and social isolation as imposed by the COVID-19 crisis. In an online questionnaire, administered in 11 countries, participants rated the relevance of wellbeing goals during the pandemic, and the effectiveness of different activities in obtaining these goals. Music was found to be the most effective activity (...) for three out of five wellbeing goals: enjoyment, venting negative emotions, and self-connection. For diversion, music was equally good as entertainment, while it was second best to create a sense of togetherness, after socialization. This result was evident across different countries and gender, with minor effects of age on specific goals, and a clear effect of the importance of music in people's lives. Cultural effects were generally small and surfaced mainly in the use of music to obtain a sense of togetherness. Interestingly, culture moderated the use of negatively valenced and nostalgic music for those higher in distress. (shrink)

J. N. Crossley [1] raised the question of whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types (C.O.T.'s). Using an earlier definition of constructive order type, A. G. Hamilton [2] presented a counterexample. Hamilton left open the general question, however, since he pointed out that Crossley considers only orderings which can be embedded in a standard dense r.e. ordering by a partial recursive function, and that his counterexample fails to (...) meet this requirement. We resolve the question by finding a C.O.T. A which meets Crossley's requirement and such that 2 + A = A but 1 + A ≠ A. At the suggestion of A. B. Manaster and A. G. Hamilton we easily extend this construction to show that for any n ≧ 2, there is a C.O.T. A such that n + A = A but m + A ≠ A for 0 < m < n. Hence, Theorem 3 of [2] and all of its corollaries hold with the new definition of C.O.T. The construction is not difficult and requires no priority argument. The techniques are similar to those developed in [3], but no outside results are needed here. (shrink)

R.I.G Hughes offers the first detailed and accessible analysis of the Hilbert-space models used in quantum theory and explains why they are so successful.

Data from various settings suggest that there is an upper limit of about four on the number of individuals who can interact in spontaneous conversation. This limit appears to be a consequence of the mechanisms of speech production and detection. There appear to be no differences between men and women in this respect, other than those introduced by women’s lighter voices.

Data on the number of adults that an individual contacts at least once a month in a set of British populations yield estimates of network sizes that correspond closely to those of the typical “sympathy group” size in humans. Men and women do not differ in their total network size, but women have more females and more kin in their networks than men do. Kin account for a significantly higher proportion of network members than would be expected by chance. The (...) number of kin in the network increases in proportion to the size of the family; as a result, people from large families have proportionately fewer non-kin in their networks, suggesting that there is either a time constraint or a cognitive constraint on network size. A small inner clique of the network functions as a support group from whom an individual is particularly likely to seek advice or assistance in time of need. Kin do not account for a significantly higher proportion of the support clique than they do for the wider network of regular social contacts for either men or women, but each sex exhibits a strong preference for members of their own sex. (shrink)

Group size is a function of relative neocortical volume in nonhuman primates. Extrapolation from this regression equation yields a predicted group size for modern humans very similar to that of certain hunter-gatherer and traditional horticulturalist societies. Groups of similar size are also found in other large-scale forms of contemporary and historical society. Among primates, the cohesion of groups is maintained by social grooming; the time devoted to social grooming is linearly related to group size among the Old World monkeys and (...) apes. To maintain the stability of the large groups characteristic of humans by grooming alone would place intolerable demands on time budgets. It is suggested that (1) the evolution of large groups in the human lineage depended on the development of a more efficient method for time-sharing the processes of social bonding and that (2) language uniquely fulfills this requirement. Data on the size of conversational and other small interacting groups of humans are in line with the predictions for the relative efficiency of conversation compared to grooming as a bonding process. Analysis of a sample of human conversations shows that about 60% of time is spent gossiping about relationships and personal experiences. It is suggested that language evolved to allow individuals to learn about the behavioural characteristics of other group members more rapidly than is possible by direct observation alone. (shrink)

Locke is the first English philosopher to be considered in this series, and that fact of itself is worthy of attention. Philosophy, of course, like science, knows no frontiers and no national boundaries. Yet it is true to say that Locke’s contribution to philosophy is typically and peculiarly English. His moderation, his emphasis upon experience, his tolerant spirit of compromise, his dislike of mystical extravagance and of metaphysical speculation, even that elusive quality of his which people call his “common sense”, (...) are English traits. His very defects, illogicality leading to inconsistency and an awkward disorderliness of thought so abhorrent to the. (shrink)

Anxiety shifts visual attention and perceptual mechanisms, preparing oneself to detect potentially threatening information more rapidly. Despite being demonstrated for threat‐related social stimuli, such as fearful expressions, it remains unexplored if these effects encompass other social cues of danger, such as aggressive gestures/actions. To this end, we recruited a total of 65 participants and asked them to identify, as quickly and accurately as possible, potentially aggressive actions depicted by an agent. By introducing and manipulating the occurrence of electric shocks, we (...) induced safe and threatening conditions. In addition, the association between electric shocks and aggression was also manipulated. Our result showed that participants have improved sensitivity, with no changes to criterion, when detecting aggressive gestures during threat compared to safe conditions. Furthermore, drift diffusion model analysis showed that under threat participants exhibited faster evidence accumulation toward the correct perceptual decision. Lastly, the relationship between threat source and aggression appeared to not impact any of the effects described above. Overall, our results indicate that the benefits gained from states of anxiety, such as increased sensitivity toward threat and greater evidence accumulation, are transposable to social stimuli capable of signaling danger other than facial expressions. (shrink)

R.I.G. Hughes presents a series of eight philosophical essays on the theoretical practices of physics. The first two essays examine these practices as they appear in physicists' treatises (e.g. Newton's Principia and Opticks ) and journal articles (by Einstein, Bohm and Pines, Aharonov and Bohm). By treating these publications as texts, Hughes casts the philosopher of science in the role of critic. This premise guides the following 6 essays which deal with various concerns of philosophy of physics such as laws, (...) disunities, models and representation, computer simulation, explanation, and the discourse of physics. (shrink)