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Concept
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
Summary
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of Monatshefte für Mathematik. Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new techniques that Gödel invented to prove the incompleteness theorems.
The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field of mathematical logic. These appear as theorems VI and XI, respectively, in the paper.
In order to prove these results, Gödel introduced a method now known as Gödel numbering. In this method, each sentence and formal proof in first-order arithmetic is assigned a particular natural number. Gödel shows that many properties of these proofs can be defined within any theory of arithmetic that is strong enough to define the primitive recursive functions. (The contemporary terminology for recursive functions and primitive recursive functions had not yet been established when the paper was published; Gödel used the word rekursiv ("recursive") for what are now known as primitive recursive functions.) The method of Gödel numbering has since become common in mathematical logic.
Because the method of Gödel numbering was novel, and to avoid any ambiguity, Gödel presented a list of 45 explicit formal definitions of primitive recursive functions and relations used to manipulate and test Gödel numbers. He used these to give an explicit definition of a formula Bew() that is true if and only if x is the Gödel number of a sentence φ and there exists a natural number that is the Gödel number of a proof of φ.
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