The non-isolating degrees are upwards dense in the computably enumerable degrees

ABSTRACT. The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degree d is isolated by a c.e. degree a if a < d is the greatest c.e. degree below d. A computably enumerable degree c is non-isolating if no d.c.e. degree above c is isolated by c. Obviously, 0 is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Arslanov et al. showed in [3] that nonzero non-isolating degrees exist and that these degrees are downwards dense in the c.e. degrees and can also occur in every jump class. In [10], Salts proved that there is an interval of computably enumerable degrees, each of which isolates a d.c.e. degree. Recently, Cenzer et al. [4] proved that such intervals are dense in the computably enumerable degrees, and hence the non-isolating degrees are nowhere dense in the computably enumerable degrees. In this paper, using a different type of construction to that of [3], we prove that the non-isolating degrees are upwards dense in the computably enumerable degrees. In the context of [4], this is the best possible such result. 1

Mind out of matter: topics in the physical foundations of consciousness and cognition

This dissertation begins with an exploration of a brand of dual aspect monism and some problems deriving from the distinction between a first person and third person point of view. I continue with an outline of one way in which the conscious experience of the subject might arise from organisational properties of a material substrate. With this picture to hand, I first examine theoretical features at the level of brain organisation which may be required to support conscious experience and then discuss what bearing some actual attributes of biological brains might have on such experience. I conclude the first half of the dissertation with comments on information processing and with artificial neural networks meant to display simple varieties of the organisational features initially described abstractly.While the first half begins with a view of conscious experience and infers downwards in the organisational hierarchy to explore neural features suggested by the view, attention in the second half shifts towards analysing low level dynamical features of material substrates and inferring upwards to possible effects on experience. There is particular emphasis on clarifying the role of chaotic dynamics, and I discuss relationships between levels of description of a cognitive system and comment on issues of complexity, computability, and predictability before returning to the topic of representation which earlier played a central part in isolating features of brain organisation which may underlie conscious experience.Some themes run throughout the dissertation, including an emphasis on understanding experience from both the first person and the third person points of view and on analysing the latter at different levels of description. Other themes include a sustained effort to integrate the picture offered here with existing empirical data and to situate current problems in the philosophy of mind within the new framework, as well as an appeal to tools from mathematics, computer science, and cognitive science to complement the more standard philosophical repertoire