Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $\omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

On the jumps of degrees below an recursively enumerable degree

We consider the set of jumps below a Turing degree, given by JB(a) = {x(1) : x <= a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high(2) r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy-at least in the form presented-by constructing pairs a(0), a(1) of distinct r.e. degrees such that JB(a(0)) = JB(a(1)) within any possible jump class {x : x' = c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity

On the strength of König's duality theorem for infinite bipartite graphs

AbstractWe prove that König's duality theorem for infinite graphs (every graph G has a matching F such that there is a selection of one vertex from each edge in F which forms a cover of G) is inherently of very high complexity in terms of both the methods of proof it requires and the computational complexity of the covers it produces. In particular, we show that there is a recursive bipartite graph such that any cover as required by the theorem is highly non-computable; indeed it must be above (in Turing degree) all the recursive iterations of the Turing jump. This implies that the theorem is proof theoretically at least as strong as the system ATR0 which is known to be strictly stronger than compactness or König's lemma. Thus the theorem cannot be proven by elementary means plus compactness. Transfinite methods are actually necessary. The actual cover given by the proof considered is seen to have an additional maximality property which makes the assertion of its existence imply a stronger system, Π11-CA0. We refine this known proof of König's theorem to show that in fact its consequences are equivalent to Π11-CA0

AN EXAMINATION OF MALE AND FEMALE DEVELOPMENT OF TRAITS THAT PLACE INDIVIDUALS AT RISK FOR ANOREXIA NERVOSA.

Anorexia nervosa has been considered to be largely a postpubescent female phenomenon. Accordingly, research in this area has focused almost exclusively on this group of females. Two overlapping perspectives have been provided as possible explanations for the prevalence of anorexia nervosa in this age group of females. These are the pubertal viewpoint (advanced by Crisp) and the sociocultural viewpoint (advanced by Garner). It was hypothesized that the effects of puberty and sociocultural propaganda would not differentiate prepubescent males or females, but that with the onset of puberty females would score higher than males on a measure of body dissatisfaction and drive for thinness. It was also predicted that with puberty well underway, even greater differences would become evident between females and males. In order to assess this hypothesis, a nonclinical sample of six hundred and thirty two prepubescent, pubescent and postpubescent elementary and high school students (two hundred and six male and four hundred and twenty four female) aged 11 to 20 years old, from schools in southwestern Ontario were given Garner and Olmsted\u27s (1984) Eating Disorder Inventory (EDI). An analysis of EDI subscale reliability (internal consistency) was performed and supported Garner\u27s contention that the scale is appropriate for administration for children as young as 11 years of age. Results indicated that females scored higher than males at all age groups on Drive for Thinness and Body Dissatisfaction. It is concluded that even as young as 11 and 12 years of age, females are experiencing dissatisfaction with body shape and are driven toward thinness to a greater extent than are males. As was predicted, as females get older, they increased in dissatisfaction with their bodily shape, however, contrary to predictions, they did not increase in their drive for thinness as they got older. Consistent with predictions, males did not increase on Drive for Thinness with age. Prepubertal males however were more dissatisfied with their body shape than were older males. These findings are consistent with the pubertal and sociocultural positions with the qualification that the sociocultural propaganda thought to play a role in the development of anorexia nervosa begins at a younger age than was commonly thought.Dept. of Psychology. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1987 .S467. Source: Dissertation Abstracts International, Volume: 48-10, Section: B, page: 3122. Thesis (Ph.D.)--University of Windsor (Canada), 1987