A Revision-Theoretic Analysis of the Arithmetical Hierarchy
Notre Dame Journal of Formal Logic 35 (2):204-218 (1994)
| Abstract | In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of definition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a different perspective on mathematically more “respectable” entities. Revision Rules were first introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they have been further developed to provide the foundations for a general theory of (possibly circular) definitions. Revision Rules are non-monotonic inductive operators that are iterated into the transfinite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then define a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,711 |
| External links |
  Try with proxy. |
| Through your library | Configure |
Similar books and articlesC. M. Asmus (2013). Vagueness and Revision Sequences. Synthese 190 (6):953-974.
Nuel Belnap (2006). Presentence, Revision, Truth, and Paradox. Philosophy and Phenomenological Research 73 (3):705–712.
Nuel D. Belnap (1982). Gupta's Rule of Revision Theory of Truth. Journal of Philosophical Logic 11 (1):103-116.
Stephen Murray Glaister (1998). Symmetry and Belief Revision. Erkenntnis 49 (1):21-56.
P. D. Welch (2003). On Revision Operators. Journal of Symbolic Logic 68 (2):689-711.
Benedikt Löwe & Philip D. Welch (2001). Set-Theoretic Absoluteness and the Revision Theory of Truth. Studia Logica 68 (1):21-41.
P. D. Welch (2001). On Gupta-Belnap Revision Theories of Truth, Kripkean Fixed Points, and the Next Stable Set. Bulletin of Symbolic Logic 7 (3):345-360.
G. Aldo Antonelli (1992). Revision Rules: An Investigation Into Non-Monotonic Inductive Definitions. Dissertation, University of Pittsburgh
Gian Aldo Antonelli (1996). What's in a Function? Synthese 107 (2):167 - 204.
Analytics
Monthly downloads |
Added to index2009-01-28Total downloads2 ( #232,722 of 551,105 )Recent downloads (6 months)1 ( #63,341 of 551,105 )How can I increase my downloads? |
My notes
Discussion
