On Gödel’s modal interpretation of the intuitionistic logic.

*(English)*Zbl 1291.03010
Béziau, Jean-Yves (ed.), Universal logic: An anthology. From Paul Hertz to Dov Gabbay. Basel: Birkhäuser (ISBN 978-3-0346-0144-3/pbk; 978-3-0346-0145-0/ebook). Studies in Universal Logic, 71-88 (2012).

Summary: The aim of this paper is to analyse Gödel’s paper “Eine Interpretation des intuitionistischen Aussagenkalküls”, presented in 1932 at the Mathematical Colloquium in Vienna and published in 1933 (reprinted and provided with an English translation in [Zbl 0592.01035]; translation reprinted in [Zbl 1290.03001]). This paper presents an interpretation from the intuitionistic propositional logic into a certain modal expansion \(\mathcal{G}\) of the classical propositional logic introduced by Gödel. We discuss Gödel’s results and several known important extensions of his conjectures. We also analyse a second paper presented at the Vienna Mathematical Colloquium in 1932 and published in 1933, in which Gödel introduces an interpretation from classical propositional logic into intuitionistic propositional logic, that he extends to the corresponding arithmetics in order to prove the relative consistency of one relative to the other. We emphasize the originality and relevance of Gödel’s results and their meaningful extensions, and analyse them under the scope of the study of interrelations between logical systems through translations between them.

For the entire collection see [Zbl 1237.03003].

For the entire collection see [Zbl 1237.03003].

##### MSC:

03B22 | Abstract deductive systems |

03-03 | History of mathematical logic and foundations |

03B20 | Subsystems of classical logic (including intuitionistic logic) |

03B45 | Modal logic (including the logic of norms) |