Godel's ProofFrom the Introduction. In 1931 there appeared in a German scientific periodical a relatively short paper with the forbidding title ""Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme"" (""On Formally Undecidable propositions of Principia Mathematica and Related Systems""). Its author was Kurt Godel, then a young mathematician of 25 at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. The paper is a milestone in the history of logic and mathematics. When Harvard University awarded Godel an honorary degree in 1952, the citation described the work as one of the most important advances in logic in modern times. At the time of its appearance, however, neither the title of Godel's paper nor its content was intelligible to most mathematicians. |
Contents
The Problem of Consistency | 8 |
Absolute Proofs of Consistency | 26 |
The Systematic Codification of Formal Logic | 37 |
An Example of a Successful Absolute Proof | 45 |
The Idea of Mapping and Its Use in Mathematics | 57 |
Gödels Proofs | 68 |
B The arithmetization of metamathematics | 76 |
The heart of Gödels argument | 85 |
Concluding Reflections | 98 |
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absolute proof algebraic arith arithmetic is consistent arithmetical calculus arithmetical formula asserts assumption axiomatic method axioms of arithmetic belongs branches of mathematics cardinal numbers consistency of arithmetic constant signs constructed contain correspond deducible derived elementary constituents elementary signs established Euclid's proof Euclidean geometry Euclidean system example finite number finitistic formal calculus formal logic formal system formalized arithmetic formula G formula with Gödel G is demonstrable given Gödel number Gödel number 13 Gödel showed Gödel's paper Gödel's Proof greatest prime Hilbert immediate successor integer mapping mathematician matical ment meta meta-mathe meta-mathematical statement metical mula notation obtained parallel axiom parallel postulate Pr(x prime greater prime number Principia Mathematica problem proof of consistency reader Richard Paradox Richardian Riemannian Riemannian geometry rules of inference sentence sentential calculus sentential connectives sentential variables sequence of formulas set of axioms sistency structure tautology tion Transformation Rules true variable with Gödel