Indeterminacy in Mathematics
Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
Assistant editors: Sam Roberts, Pawel Pawlowski
About this topic
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 1999. For an in-depth study of arithmetical incompletness, see Franzén 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. |
Show all references
Related categories
Siblings:
- Mathematical Fictionalism (92)
- Mathematical Nominalism (220)
- Mathematical Platonism (441)
- Mathematical Psychologism (25)
- Mathematical Structuralism (324)
- Mathematical Neo-Fregeanism (375)
- Debunking Arguments about Mathematics (12)
- Indispensability Arguments in Mathematics (237)
- Numbers (372)
- The Nature of Sets (258 | 98)
Jobs in this area
Lecturer in Applied Ethics
Term (visiting) Assistant Professor
Phyllis W. Nicholas Director of The Janet Prindle Institute for Ethics
Jobs from PhilJobs
27 found
Order:
1 filter applied
|
Off-campus access
Using PhilPapers from home?
Create an account to enable off-campus access through your institution's proxy server. Monitor this page
Be alerted of all new items appearing on this page. Choose how you want to monitor it:
Editorial team
General Editors:
David Bourget (Western Ontario) David Chalmers (ANU, NYU) Area Editors: David Bourget Gwen Bradford Berit Brogaard Margaret Cameron David Chalmers James Chase Rafael De Clercq Ezio Di Nucci Barry Hallen Hans Halvorson Jonathan Ichikawa Michelle Kosch Øystein Linnebo JeeLoo Liu Paul Livingston Brandon Look Manolo Martínez Matthew McGrath Michiru Nagatsu Susana Nuccetelli Giuseppe Primiero Jack Alan Reynolds Darrell P. Rowbottom Aleksandra Samonek Constantine Sandis Howard Sankey Jonathan Schaffer Thomas Senor Robin Smith Daniel Star Jussi Suikkanen Lynne Tirrell Aness Kim Webster Other editors Contact us Learn more about PhilPapers |