We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show (...) that for all n ≥ 2, there exists a Δ 0 n+1 n-cohesive set. We improve this result for n = 2 by showing that there is a Π 0 2 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0 (n+1) and also characterize the jumps of the n-r-cohesive degrees. (shrink)

The consistency strength of the ∑2 priority method is I∑2, yet classical theorems proven by this method have been proved from I∑1. Is there a statement about the structure of the r.e. degrees that can be proved using a ∑2 argument and cannot be proved from I∑1?We rule out statements in the language of partial orderings of the form …[], where is quantifier-free, by showing that the following can be proved in I∑1.If P is any recursive partial ordering with a (...) maximal point d, and a is any nonrecursive incomplete r.e. degree, then P can be embedded into the r.e. degrees by an embedding sending d to a. (shrink)

The understanding of food cue associated neural activations that predict future weight variability may guide the design of effective prevention programs and treatments for overeating and obesity. The current study investigated the association between brain response to different food odors with varied energy density and individual changes of body mass index (BMI) over two years. Twenty-five participants received high-fat (chocolate and peanut), low-fat (bread and peach) food odors and a nonfood odor (rose) while the brain activation was measured using functional (...) magnetic resonance imaging (fMRI). BMIs were calculated with participant’s self-reported body weight and height collected at the time of fMRI scan and again at two years later. Regression analyses revealed significant negative correlations between BMI increase over 2 years and brain activation of the bilateral precuneus and the right posterior cingulate cortex in response to high-fat versus low-fat food odors. In addition, brain activation of the right supplementary motor area in response to food vs non-food odor was negatively correlated to subsequent BMI increase over 2 years. Taken together, the current findings suggest that individual differences in neural responsivity to (high calorie) food odors in brain regions of the default mode and motor control network serve as a neural marker for future BMI change. (shrink)

It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible in (...) various senses. (shrink)

Smell perception plays an important role in eating behavior and might be involved in body weight gain. Since a body of literature implies that olfactory perception and function is hampered in obesity, we here investigate neuroanatomical correlates of this phenomenon. We assessed olfactory bulb volume with magnetic resonance imaging in 67 healthy participants with a body mass index from 18.9 to 45.4 kg/m2. Moreover, we obtained psychophysiological data on olfactory ability and self-report measurements on eating behavior. Additionally, we collected parameters (...) associated with metabolic health in obesity to investigate recently proposed mechanistic explanatory models of why olfaction may be altered in obesity. We showed that OB volume was significantly lower in participants with obesity when compared to those of normal weight. Moreover, we found weak to moderate negative correlations between OB volume and BMI and related measures of metabolic health, especially leptin, body fat percentage, waist-height ratio and insulin resistance. However, neither OB volume nor BMI were related to olfactory function in our young and healthy sample. Nevertheless, our results provide first indications that obesity is associated with brain anatomical changes in the OBs. (shrink)