SafekipediaOn Formally Undecidable Propositions of Principia Mathematica and Related Systems
Adapted from Wikipedia · Discoverer experience
This article is about the paper. For theorems proved in this paper, see Gödel's incompleteness theorems.
"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 und Physik. 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.
Outline and key results
This paper by Kurt Gödel introduced important ideas in mathematical logic. Gödel showed that in any system of math that is strong enough, there are statements that cannot be proven true or false. He did this by giving each math sentence a special number, now called Gödel numbering.
He also used sentences that refer to themselves, like one that says "This statement cannot be proven." This helped him show that some truths in math cannot be proven using the rules of the system. These ideas changed how we understand math and logic forever.
Main article: incompleteness theorems
Main articles: Gödel's incompleteness theorems and Gödel numbering
Further information: List of topics related to Gödel's incompleteness theorems
Published English translations
Three English translations of Kurt Gödel's important paper were published during his lifetime. The first, translated by Bernard Meltzer, came out in 1963. Later, Elliott Mendelson made another translation for a book called The Undecidable. Finally, Jean van Heijenoort translated the paper for a collection named From Frege to Gödel. Each translation had its own challenges and reviews, but they all helped make Gödel's ideas accessible to more people. There is also a version based on lectures Gödel gave in 1934, transcribed by Stephen Kleene and J. Barkley Rosser.
Main articles: The Undecidable, From Frege to Gödel: A Source Book in Mathematical Logic
This article is a child-friendly adaptation of the Wikipedia article on On Formally Undecidable Propositions of Principia Mathematica and Related Systems, available under CC BY-SA 4.0.