„All consistent axiomatic formulations of number theory include undecidable propositions. […] If consistency is the minimal condition under which symbols acquire passive meanings, then its complementary notion, completeness, is the maximal confirmation of those passive meanings. Where consistency is the property that ‚Everything produced by the system is true‘, completeness is the other way round: ‚Every true statement is produced by the system‘. […] Gödel’s Incompleteness Theorem says that any system which is ’sufficiently powerful‘ is, by virtue of its power, incomplete, in the sense that there are well-formed strings which express true statements of number theory, but which are not theorems. (There are truths belonging to number theory which are not provable within the system.)“ (p. 17, 100f [pass.]) #Hofstadter #Gödel #Escher #Bach #IncompletenessTheorem #NumberTheory
Schlagwort-Archive: Hofstadter
Theories on the Origin of Life
Zitat
„There are various theories on the origin of life. They all run aground on this most central questions: »How did the Genetic Code, along with the mechanisms for its translation (ribosomes and tRNA melecules), originate?«“ (p. 548) #Hofstadter #OriginOfLife #GeneticCode
Process of Reasoning
Zitat
„We use the word »all« in a few ways which are defined by the thought processes of reasoning. That is, there are rules which our usage of »all« obeys. We may be unconscious of them, and tend to claim we operate on the basis of the meaning of the word; but that, after all, is only a circumlocution for saying that we are guided by rules which we never make explicit.“ (p. 60) #Hofstadter #rules #meaning #ProcessOfReasoning
Hofstadter: Incompleteness Theorem
Zitat
„The proof of Gödel’s Incompleteness Theorem hinges upon the writing of a self-referential mathematical statement, in the same way as the Epimenides paradox is a self-referential statement of language.“ (p. 17) #Hofstadter #Gödel #Epimenides #IncompletenessTheorem #Self-Referential
Real-World Thinking
Zitat
„Real-world thinking is quite different from what happens when we do a multiplication of two numbers“.
Hofstadter, Douglas R., Gödel, Escher, Bach: An Eternal Golden Braid. A Metaphorical Fugue on Minds and Machines in the Spirit of Lewis Carroll. New York, 1979, p. 569