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Unformatted text preview: CS103A HO #56 Gödel's Incompleteness Theorem 3/10/08 1 Gödel's Incompleteness Theorem Kurt Gödel (1906 – 1978) Gödel, Kurt (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physic , 38, 173-198. On formally undecidable propositions of Principia Mathematica and related systems I References: Gödel's Proof by Ernest Nagel, James R. Newman, Douglas R. Hofstadter (Editor) NYU Press; Revised edition, 2001 Wikipedia Gödel Gödel's incompleteness theorems An Introduction to Gödel's Theorems by Peter Smith Cambridge University Press, 2007 Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén A K Peters, 2005 God Created the Integers by Stephen Hawking (Editor) (contains Gödel's paper) Running Press, 2007 Mathematics is deductive, not experimental The axiomatic method and the notion of logical proof goes back to the Greeks--A small number of axioms and some rules of inference are used to produce all other theorems in the system It was hoped that adequate systems of axioms could be found for each area of mathematics. For example, Euclid's axioms of geometry, or Peano's axioms of arithmetic. Gödel showed that this could not be done. There are inherent limitations in such systems. Let's set the stage... Mathematical Progress in the 19 th Century The Greeks had proposed three problems in geometry that had not been solved for 2000 years: --to trisect any angle with a compass and straight edge--to construct a cube with a volume twice the volume of a given cube--to construct a square equal in area to that of a given circle. In the 19th century it was proved that these constructions are logically impossible. Mathematical Progress in the 19 th Century Even more important was another problem: Euclid's fifth axiom is equivalent to the assumption that through a point outside a given line only one parallel line can be drawn. Unlike the other axioms, this did not seem self-evident to the Greeks. They tried, unsuccessfully, to deduce it from the other for axioms. In the 19 th century, the impossibility of deducing the parallel axiom from the others was demonstrated. This was important because--it showed that it was possible to prove the impossibility of proving certain propositions within a given system--it showed that there was more to geometry than Euclid CS103A HO #56 Gödel's Incompleteness Theorem 3/10/08 2 Mathematical Progress in the 19 th Century The emergence of non-Euclidean geometries, where Euclid's axioms were not true, raised interesting questions:--Were the axioms for the non-Euclidean geometries consistent? (How could you prove that they didn’t lead to contradictory theorems?)--Were the Euclidean axioms consistent?...
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- Winter '07
- Computer Science, Mathematical logic, Axiom, Proof theory, Kurt Gödel