This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

Generalized Recursion Theory

Summary 'Topics in Programming Languages' explores the arch from the formation of alphabet and classical philosophy to artificial programming languages in the structure of one argumentative topics list: as if it were philosophy interpreted and programmed. One such endeavour is taken to tend toward phonetics and sounds of speech analysis with λ-calculus, and, ultimately, Prolog - the programming language of choice in artificial intelligence - born of the natural language processing reverie and delusion. The well-ordered list of arguments targets the conceptual tree behind both the functional and the logical, the procedural and the declarative paradigms in programming languages by studying close the ascendum (convolution) of the Aristotelian efficient cause into the notions of function (Leibniz), rule (Kant) and algorithm as effective procedures in computation (Church-Turing). The Author Luís Manuel Cabrita Pais Homem graduated in Philosophy in the Faculty of Letters of the University of Lisbon in 2005. He concluded the Master in the same He is currently completing his doctoral thesis. the Post-Graduate Program holds a Quality Grant, taking in automatic passage to Doctorate, the author is currently preparing the PhD thesis subordinated to the same theme. The author is an integrated member of the Centre for Philosophy of Science of the University of Lisbon since the summer of 2011. Readership Scholars, students, programmers, computer scientists Contents Section I - Arguments; α) The phonetics and philosophical argument; β) The symbolic or rational argument; γ) The difficulty argument; δ) The content-and-form artificial intelligence argument; ε) The efficient cause argument; ζ) The model theory argument; Notes Section II - Arguments; The endogenous to exogenous language argument; θ) The efficient cause continuance argument; ι) The reviewing incommensurability argument; κ) The functional and declarative programming languages argument; Notes Section III - Arguments; λ) The λ-calculus argument; μ) The Prolog argument Notes Section IV - Topics in programming languages: a philosophical analysis through the case of prolog; Summary; State of the art; Goal; Detailed description Bibliography

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

This volume honors Professor Andrzej Grzegorczyk, the nestor of Polish logicians, on his 85th anniversary. The editors would like to express the respect and sympathy they have for him. His textbook The Outline of Mathematical Logic has been published in many editions and translated into several languages. It was this textbook that introduced many of us into the world of mathematical logic. Professor Grzegorczyk has made fundamental contributions to logic and to philosophy. His results, mainly on hierarchy of primitive recursive functions, known as the Grzegorczyk hierarchy, are of fundamental importance to theoretical computer science. In particular, they were precursory for the computational complexity theory. The editors would like to stress that this special publication celebrates a scientist who is still actively pursuing genuinely innovative directions of research. Quite recently, Andrzej Grzegorczyk gave a new proof of undecidability of the first order functional calculus. His proof does not use the arithmetization of Kurt Gödel. In recognition of his merits, the University of Clermont-Ferrand conferred to Professor Andrzej Grzegorczyk the Doctorat Honoris Causa. The work and life of Professor Andrzej Grzegorczyk is presented in the article by Professors Stanislaw Krajewski and Jan Wolenski. The papers in this collection have been submitted on invitational basis.