This paper presents hitherto unpublished writings of Kurt Gödel concerning logical, epistemological, theological, and physical antinomies, which he generally considered as “the most interesting facts in modern logic,” and which he used as a basis for his famous metamathematical results. After investigating different perspectives on the notion of the logical structure of the antinomies and presenting two “antinomies of the intensional,” a new kind of paradox closely related to Gödel’s ontological proof for the existence of God is introduced and completed (...) by a compilation of further theological antinomies. Finally, after a presentation of unpublished general philosophical remarks concerning the antinomies, Gödel’s type-theoretic variant of Leibniz’Monadology, discovered in his notes on the foundations of quantum mechanics, is examined. Most of the material presented here has been transcribed from the Gabelsberger shorthand system for the first time. (shrink)

Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order (...) modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof. (shrink)

This paper takes a closer look at the phenomenon of the ineffability of God from a purely logical perspective. In doing so, it pursues two main objectives. First, as to this day many philosophers speak – without hesitation – of the ‘paradox’ of ineffability closely associated with Liar-like sentences, it clarifies the situation by showing that ineffability is by no means paradoxical in the strict logical sense. Secondly, it uses a new information-theoretic approach in order to clearly distinguish between what (...) is called ‘substantial’ and ‘non-substantial’ predicates, a distinction which finally allows for a rendering of the dogma of ineffability which does not entail contradiction. (shrink)

In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of a (...) computing machine which closely resembles Turing's groundbreaking conception. After introducing the notebooks on quantum mechanics, this article describes Gödel's vitalistic Weltbild and the ideas leading to the development of his computing machine. It investigates a notion of lawlike sequence which closely resembles Turing's concept of a computable number and which Gödel himself calls ‘problematic’, and compares it to the opposed concept of randomness, drawing upon the notion of program-size complexity. Finally, Gödel's machine is implemented in a dialect of the Lisp programing language. (shrink)

This paper analyses an amazingly close analogy between models of generalised trinitarin logics on the one hand side and class hierarchies in the field of object-oriented programming on the other, thus linking philosophy of religion and computer science. In order to bring out this analogy as clear and precise as possible, we utilise a metaobject protocol for the actual implementation of the theological models. These formal implementations lead to the insight that the analogy can be pushed even further, and we (...) lay bare and analyse the close relation between the theological notion of subordination of divine persons and precedence in structures of multiple inheritance. The implementation of theoretical godheads finally leads to new metaobject programming techniques, thus underlining the cross-fertilisation between theology and computer science. (shrink)

This paper collects and presents unpublished notes of Kurt Gödel concerning the field of many-valued logic. In order to get a picture as complete as possible, both formal and philosophical notes, transcribed from the Gabelsberger shorthand system, are included.

This book provides detailed transcriptions of two notebooks written by Kurt Gödel in Vienna in 1937/38 in the nearly forgotten Gabelsberger shorthand system. The first of these notebooks, simply entitled as the Protokoll-book, contains notes on conversations Gödel had with people like Rudolf Carnap, Rose Rand, Friedrich Waismann, Else Frenkel-Brunswik, and many others who were—at least to some degree—connected to the Vienna Circle. It also covers detailed descriptions of the regular meetings organized by Edgar Zilsel. The second notebook includes notes (...) on a series of lectures given at the Vienna Psychological Institute, which was led by Karl Bühler at the time. Both notebooks are part of Gödel’s huge Nachlass kept at the Institute for Advanced Study in Princeton, which consists of literally thousands of stenographic pages covering logic and the foundations of mathematics, philosophy, physics, and theology. The now transcribed and commented notes reveal a very personal side of Gödel which has—to a large extent—been unknown to a mainly scientific-oriented audience. The book is of interest to people wanting to learn about Gödel’s personal background in Vienna in the late 1930s as well as his keen interest in philosophy, psychology, and parapsychology. (shrink)