Economics Dictionary of ArgumentsHome
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| Kurt Gödel: Kurt Gödel (1906 – 1978) was a logician, mathematician, and philosopher. He is best known for his incompleteness theorems, which show that within any axiomatic system powerful enough to express basic arithmetic, there will always be statements that can neither be proven nor disproven within that system. Major works are "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" (1931), "Consistency-Proof for the Generally Covariant Gravitational Field Equations" (1939), "What is Cantor's Continuum Problem?" (1947), "Russell's Mathematical Logic" (1951), "On Undecidable Propositions of Formal Mathematical Systems" (1956). See also Incompleteness, Completeness, Proofs, Provability.
_____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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Hennig Genz on Goedel - Dictionary of Arguments
II 213 Goedel/incompleteness/Hilbert/Genz: in 1917 Hilbert had drawn up the program to summarize all mathematics in a scheme in the logic of the 1st level, Goedel proved in 1931 that this was not possible. It works well for Euclidean and non-Euclidean geometry, but not for addition and multiplication, if you take their derivation rules together. It is always about sentences that are formulated in a language, but cannot be derived or refuted. >Incompleteness. Abundance/Genz: in poor languages, all statements that can be formulated in them can either be derived or disproved. The richer they are, the more statements can be formulated that do not succeed. >Semantic closure. II 214 These sentences make a statement about themselves, namely that they cannot be derived. Solution: a solution is the extension of the language. For example, to accept his negation as an axiom. >Extension, >Levels (order), >Description levels. Problem: in every extension there are new non-derivable sentences. Deductibility: a language in which any sensible phrase at all could be derived would allow to derive contradictions. >Derivation, >Derivability. >Contradictions._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Gz I H. Genz Gedankenexperimente Weinheim 1999 Gz II Henning Genz Wie die Naturgesetze Wirklichkeit schaffen. Über Physik und Realität München 2002 |
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