Colegas: Gostaria de divulgar aqui um livro talvez pouco conhecido, mas que traz uma excelente visão sobre a interpretação funcional da aritmética, análise e teoria dos conjuntos. O livro revisita a famosa interpretação dialética de Gödel da aritmética de Heyting e sua generalização para tipos finitos, e expõe muito bem este temas, incluindo a interpretação de Diller-Nahm na aritmética de Heyting e Peano em tipos finitos e outros temas clássicos.
Justus Diller foi meu "Gastgeber" (anfitrião) no tempo que passei em Münster como bolsista da Humboldt Stiftung, é um excelente lógico, matemático e filósofo da matemática e da lógica (embora talvez ele nem pretenda isso...). Foi também visitante no CLE Unicamp. Aprendi muito com ele. Recomendo realmente este livro. Abs, Walter [Inglês abaixo, c/c Diller] =================== Dear Colleagues: I would like to share here a book that is perhaps little known, but which provides an excellent insight into functional interpretation of arithmetic, analysis and set theory. The book revisits the famous Gödel's dialectical interpretation of Heyting arithmetic and its generalization to finite types, and exposes these very well, including the Diller-Nahm's interpretation of Heyting and Peano arithmetic on finite types, and other classical topics. Justus Diller was my "Gastgeber" (host) during my time in Münster on a Humboldt Stiftung scholarship, is an excellent logician, mathematician and philosopher of mathematics and logic (although perhaps he doesn't even intend to...). He was also a visitor at CLE Unicamp. I learned a lot from him. I really recommend this book. Best, Walter ================================== "Functional Interpretations: From the Dialectica Interpretation to Functional Interpretations of Analysis and Set Theory " Justus Diller (Univ of Münster, Germany) World Scientific Pub, 2019 https://www.worldofbooks.com/en-au/books/justus-diller-univ-of-munster/functional-interpretations-from-the-dialectica-interpretation-to-functional-inte/9789814551397 This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Goedel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, in particular the Dialectica interpretation and its generalization to finite types, the Diller-Nahm interpretation, are studied on Heyting as well as Peano arithmetic in finite types and extended to functional interpretations of constructive as well as classical systems of analysis and set theory. Kreisel's modified realization and Troelstra's hybrids of it are presented as interpretations of Heyting arithmetic and extended to constructive set theory, both in finite types. They serve as background for the construction of hybrids of the Diller-Nahm interpretation of Heyting arithmetic and constructive set theory, again in finite types. All these functional interpretations yield relative consistency results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations. Table of Contents Arithmetic: Primitive Recursive Functionals; - (Diller - Nahm) Interpretation of Heyting Arithmetic in Finite Types; The Dialectica Interpretation and Equality Functionals; Simultaneous Recursions in Linear Types; Computability, Consistency, Continuity; Modified Realization and its Hybrids; Hybrids of the -Interpretation; N-Interpretations; Interpretations of Classical Arithmetic; Extensionality and Majorizability; Analysis: Bar Recursive Functionals; - and Dialectica Interpretation of Bar Induction by Bar Recursion; Functional Interpretations of Classical Analysis; Computability of Bar Recursive Functionals; Set Theory: Constructive Set Functionals; Kripke - Platek Set Theory and Its Functional Interpretations; Constructive Set Theory and Its -Interpretation; Modified Realizations of Constructive Set Theory; The Q-Hybrid of the -Interpretation of Constructive Set Theory in Finite Types; Majorizability of Constructive Set Functionals. =========================== Walter Carnielli, Professor Centre for Logic, Epistemology and the History of Science and Department of Philosophy University of Campinas –UNICAMP 13083-859 Campinas -SP, Brazil Phone: (+55) (19) 3521-6517 Institutional e-mail: walter.carnie...@cle.unicamp.br Website: http://www.cle.unicamp.br/prof/carnielli -- Você está recebendo esta mensagem porque se inscreveu no grupo "LOGICA-L" dos Grupos do Google. Para cancelar inscrição nesse grupo e parar de receber e-mails dele, envie um e-mail para logica-l+unsubscr...@dimap.ufrn.br. Para ver esta discussão na web, acesse https://groups.google.com/a/dimap.ufrn.br/d/msgid/logica-l/CAOrCsLcH_gynoLchVEaTjiCzFNsorDcOVGwMJerKXw_f1A1nNQ%40mail.gmail.com.
