In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.
In the spring of , Kurt Gödel held a lecture course on intuitionistic logic at the Institute for Advanced Study in Princeton. Two spiral notebooks labelled simply "Vorl." and two sets of loose notes contain handwritten notes for the lecture course. The lecture notes divide into two themes. The rst part is an introduction to intuitionistic logic. The second part is a detailed presentation of Gödel's functional interpretation of Heyting Arithmetic and its applications.The general aim of the lectures is to examine the constructivity of intuitionistic logic. In the rst part of the lectures, Gödel focuses heavily on the interconnection between intuitionistic and classical logic. The standard proof explanation of the intuitionistic logic was, he believed, not adequate to show the constructive character of intuitionistic logic. By reinterpreting intuitionistic logic in a more precise way, Gödel wants to prove that Heyting Arithmetic is properly constructive in the sense that it has the existence property. This reinterpretation is Gödel's functional system Σ, and the Princeton course is the most detailed presentation of it.The theme of the lectures was closely connected to Gödel's previous talks of and , as well as a lecture given at Yale University in April . In the lecture "The present situation in the foundations of mathematics" given in Cambridge, Massachusetts, in , Gödel argues that intuitionistic logic is not an ideal basis for a constructive foundation of mathematics because of the nature of its logical operations and the proof explanation. In his "Zilsel lecture" of , he mentions an alternative interpretation of the logical operations in terms of a system of primitive recursive functionals of higher types. Finally, the system is developed in detail in the Princeton course and the Yale lecture. These results -apart from the Princeton lectures -were published posthumously in Gödel's Collected Works in ; the rst published article on the functional interpretation appeared years after the Princeton course, in the journal Dialectica in .In what follows, I will give an overview of the lecture course, highlighting the features which are missing from the other works of the s and early s. Apart from higher level of detail, the new aspects include an alternative version of Gödel's negative translation between Peano and Heyting Arithmetic (Gödel b), the "truth table theorem" that proves that classical and intuitionistic
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.