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Gödel, Kurt

system mathematics logic consistency

[goe dl] (1906–78) Austrian–US mathematician: showed that mathematics could not be totally complete and totally consistent.

Growing up, Gödel was frequently ill and had a life-long concern with his health. He studied mathematics at Vienna, saw much of the development of the positivist school of philosophy and was apparently unconvinced. In 1930 he received his PhD, proving in his thesis that first-order logic is complete–so that in first-order logic every statement is provable or disprovable within the system. He then investigated the larger logical system put forward by B Russell (1872–1970) and A N Whitehead (1861–1947) in their Principia mathematica , and his resulting paper of 1931 may well be the most significant event in 20th-c mathematics.

The paper was titled ‘On Formally Undecidable Propositions of Principia mathematica and Related Systems’ and showed that arithmetic was incomplete. In any consistent formal system able to describe simple arithmetic there are propositions that can be neither proved nor disproved on the basis of the system. Gödel also showed that the consistency of a mathematical system such as arithmetic cannot necessarily be proved within that system. Thus a larger system may have to be used to prove consistency, and its consistency assumed; all pretty unsatisfactory. The programme for developing mathematical logic suggested by , F L G Frege (1848–1925) and Russell was therefore untenable, and it is now clear that there is no set of logical statements from which all mathematics can be derived.

Between 1938 and 1940 Gödel showed that restricted set theory cannot be used to disprove the axioms of choice or the continuum hypothesis; this was extended in 1963 when P J Cohen showed that they are independent of set theory. Gödel also contributed to general relativity theory and cosmology, and was a close friend of at Princeton. Gödel had married and emigrated to Princeton in 1938 when the Nazis took Austria, and was a professor there from 1953–76. He was an unassuming man with a variety of interests.

Göle, Nilüfer (1953–) - PERSONAL HISTORY, INFLUENCES AND CONTRIBUTIONS, BIOGRAPHICAL HIGHLIGHTS, PERSONAL CHRONOLOGY:, THE WORLD’S PERSPECTIVE, LEGACY [next] [back] Fuzzy Techniques for Context Representation - Context Representation Formalisms, Fuzzy Context: Formalized Reality, Fuzzy Ontologies

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