In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz or Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallelisms between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a (...) science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.

There is no need to introduce Gottfried Wilhelm Leibniz, a great philosopher, theologian, diplomat, creator of the infinitesimal calculus and founder of the Academy of Sciences in Berlin. He also planned the development of the so-called Lingua characteristica. Literally taken, the name of the language means a language of letters, a graphic language, also called a characteristica universalis. It was meant to be a way of expressing meanings, as modeled after methods used in arithmetic and geometry and having unusual properties.

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

We present an abstract framework in which we give simple proofs for Gödel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’, Chaitin’s and Kritchman-Raz’s Theorems.

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

This text is expository. We explain Gödel’s ‘Master Argument’ for incompleteness as distinguished from the 'official' proof of his 1931 paper, highlight its attractions and limitations, and explain how some of the limitations may be transcended by putting it in a more abstract form that makes no reference to truth.

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...) and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.

Contacts of the two logicians are listed, and all Gödel's written mentions of Tarski's work are quoted. Why did Gödel almost never mention Tarski's definition of truth in his notes and papers? This puzzle of Gödel's silence, proposed by Feferman, is not merely biographical or psychological but has interesting connections to Gödel's philosophical views.No satisfactory answer is given by the three “standard” explanations: no need to repeat the work already done; Tarski's achievement was obvious to Gödel; Gödel's exceptional caution. In (...) fact, , Tarski had done the work, but Gödel almost never mentioned the achievement; , the obviousness is no explanation for the omission of Tarski's work in contexts in which an application of the definition of satisfaction was useful, and even necessary; , the point was not just caution: if Gödel had felt the need to mention the program of scientific semantics he could easily have done that in his manuscripts, or in conversations.Three ideas, detectable in Gödel's approach, can help us understand Gödel's silence: the idea of truth as the intuitive provability in the most general sense; defining it set-theoretically would contribute nothing. the idea of truth as an inexhaustible idea in the sense of Kant; “truth in general” is a category that must be applicable to all kinds of sentential expressions; also, while for Gödel language was secondary, Tarski's definition is focused on language. the idea of logic as the universal language, in Hintikka's sense, as opposed to the perception of logic as a reinterpretable calculus; hence the thesis that semantics is inexpressible. Gödel always remains a Platonist who asks a natural question: what does really happen in the realm of abstract objects? (shrink)