There exists a countable structure of Scott rank where and where the -theory of is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its -theory. Most well-known models with unbounded atoms below also realize a non-principal -type; such a model that preserves the Σ1-admissibility of will have Scott rank . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 301–318. [4]] produces a hyperarithmetical model of Scott (...) rank whose -theory is ω-categorical. A computable variant of Makkai’s example is produced in [W. Calvert, S.S. Goncharov, J.F. Knight, J. Millar, Categoricity of computable infinitary theories, Arch. Math. Logic . [1]; J. Knight, J. Millar, Computable structures of rank J. Math. Logic . [2]]. (shrink)

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years.

We show, however, that this is not always the case.

Georg Kreisel suggested various ways out of the Gödel incompleteness theorems. His remarks on ways out were somewhat parenthetical, and suggestive. He did not develop them in subsequent papers. One aim of this paper is not to develop those remarks, but to show how the basic idea that they express can be used to reason about the Lucas-Penrose-Putnam arguments that human minds are not finitary computational machines. Another aim is to show how one of Putnam’s two anti-functionalist arguments avoids the (...) logical error in the Lucas-Penrose arguments, extends those arguments, but succumbs to an absurdity. A third aim is to provide a categorization of the Lucas-Penrose-Putnam anti-functionalist arguments. (shrink)

We introduce a quite natural Frege-style set theory, which we call Strong-Frege-2 equation image, a sort of simplification of the theory considered in 13 and 1 . We give a model of a weaker variant of equation image, called equation image, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π11 sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of (...) 5 and 15. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative (...) conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles. (shrink)

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the (...) algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory. (shrink)

Sacks [23] asks if the metarecursively enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as [Formula: see text] or, equivalently, that of the truth set of [Formula: see text].