Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.
This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages.
The book also examines the underlying theory and principles of computation and is highly suitable to the undergraduate courses in computer science and information technology.
A distinctive feature of this text is its gentle and gradual introduction of the necessary mathematical tools in the context in which they are used.
Written for prospective computer scientist, computer engineers, or applied mathematicians, who want to learn about the ideas that inspire computer science, this edition contains an extensive coverage of logic, setting it apart from similar ...
This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. aThe text includes both the standard material for a first course in ...
Computability: Computable Functions, Logic, and the Foundations of Mathematics
Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.
Automata, Computability and Complexity: Theory and Applications
Automata, Computability and Complexity: Theory and Applications
A comprehensive introduction to eight major approaches to computation on uncountable mathematical domains.