Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
|Summary||A sentence C is independent of a theory T iff neither C, nor the negation of C is derivable from T. A theory is negation-complete iff no sentence in its language is independent of it. Some of key results in metamathematics are independence theorems. According to arithmetical incompleteness theorem, no consistent (recursively axiomatizable) extension of a relatively weak arithmetic is negation-complete. Another important independence result is the independence of the Conituum Hypothesis of the axioms of standard set theory. (There are numerous other examples in analysis, combinatorics, group theory and set theory.) Independence results seem to have impact on philosophical views on mathematical truth and mathematical knowledge. Are sentences independent of mainstream theories determinately true or false and why? If yes, how can we know, which is it? If no, what philosophical views about mathematics are consistent with this view and how are they motivated?|
|Key works||Gödel 1931, Gödel 1940, Gödel 1947, .Cohen 1963, Feferman manuscript and Feferman et al 2000. For an in-depth study of arithmetical incompletness, see Franzen 2003.|
|Introductions||A great introduction to arithmetical incompleteness theorems and related issues is Smith 2007. A more advanced book is Lindstrom 2002. Franzén 2005 is invaluable. See also Feferman manuscript and Feferman manuscript. As for set-theoretic indeterminacy, see Koellner 2010 and references therein.|
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