Functional Interpretations
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Walter Carnielli
Jul 31, 2021, 11:28:28 AM7/31/21
to Lista acadêmica brasileira dos profissionais e estudantes da área de LOGICA, Justus Diller
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.c...@cle.unicamp.br
Website: http://www.cle.unicamp.br/prof/carnielli
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.c...@cle.unicamp.br
Website: http://www.cle.unicamp.br/prof/carnielli
Valeria de Paiva
Aug 1, 2021, 3:19:12 AM8/1/21
to Walter Carnielli, Lista acadêmica brasileira dos profissionais e estudantes da área de LOGICA, Justus Diller
Pois e' Walter, a minha tese de doutorado fala justamente da Interpretacao Dialectica, faz uma versao categorica e usa a variante de J. Diller e Nahm pra mostrar como voltar pra logica intuicionista.
A tese esta' em https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf, a parte sobre Dialectica e Diller-Nahm num artigo em http://www.cs.bham.ac.uk/~vdp/publications/dial87.pdf.
abracos,
Valeria
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Walter Carnielli
Aug 1, 2021, 11:04:48 AM8/1/21
to Valeria de Paiva, Lista acadêmica brasileira dos profissionais e estudantes da área de LOGICA, Justus Diller
Ola Valéria,
Muito bacana, vou dar uma olhada, e passo a ele. Vai ficar muito satisfeito.
Abraços,.
Walter
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