Article ID: CBB001213927

Completeness: From Gödel to Henkin (2014)


This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödel's and Tarski's crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity in the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödel's proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkin's work concerning the generalization of his completeness proof to any logic from his initial work in type theory.

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Authors & Contributors
Gödel, Kurt
Dawson, John W., Jr.
Feferman, Solomon
Heijenoort, Jean van
Solovay, Robert M.
Parsons, Charles D.
History and Philosophy of Logic
Studies in History and Philosophy of Science
Perspectives on Science
The Review of Modern Logic
Berichte zur Wissenschaftsgeschichte
Oxford University Press
Cambridge University Press
Princeton University Press
Harvard University Press
de Gruyter
Philosophy of mathematics
Incompleteness theorems
Philosophy of science
Gödel, Kurt
Hilbert, David
Russell, Bertrand Arthur William
Cantor, Georg Ferdinand Ludwig
Frege, Gottlob
Von Neumann, John
Time Periods
20th century
19th century
20th century, early
21st century
Middle and Near East

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