∑1 definitions with parameters
Journal of Symbolic Logic 51 (2):453 - 461 (1986)
| Abstract | Let p be a set. A function φ is uniformly σ 1 (p) in every admissible set if there is a σ 1 formula φ in the parameter p so that φ defines φ in every σ 1 -admissible set which includes p. A theorem of Van de Wiele states that if φ is a total function from sets to sets then φ is uniformly σ 1R in every admissible set if anly only if it is E-recursive. A function is ES p -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ES p - recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable than a total function on sets is ES p -recursive if and only if it is uniformly σ 1 (p) in every admissible set. b) For any p, if φ is a function on the ordinal numbers then φ is ES p -recursive if and only if it is uniformly ∑ 1 (p) in every admissible set | |||||||||
| 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: 5,709 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesMichael Rathjen (1992). A Proof-Theoretic Characterization of the Primitive Recursive Set Functions. Journal of Symbolic Logic 57 (3):954-969.
Robert E. Byerly (1982). An Invariance Notion in Recursion Theory. Journal of Symbolic Logic 47 (1):48-66.
Mark Nadel & Jonathan Stavi (1977). The Pure Part of HYP(M). Journal of Symbolic Logic 42 (1):33-46.
Steffen Lempp & Theodore A. Slaman (1989). A Limit on Relative Genericity in the Recursively Enumerable Sets. Journal of Symbolic Logic 54 (2):376-395.
Shaughan Lavine (1992). A Spector-Gandy Theorem for cPCd(A) Classes. Journal of Symbolic Logic 57 (2):478 - 500.
Fred G. Abramson (1979). Σ1-Separation. Journal of Symbolic Logic 44 (3):374 - 382.
H. R. Strong (1970). Construction of Models for Algebraically Generalized Recursive Function Theory. Journal of Symbolic Logic 35 (3):401-409.
Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (01):91-112.
Sy D. Friedman (1979). HC of an Admissible Set. Journal of Symbolic Logic 44 (1):95-102.
Analytics
Monthly downloads |
Added to index2009-01-28Total downloads5 ( #160,518 of 549,699 )Recent downloads (6 months)1 ( #63,425 of 549,699 )How can I increase my downloads? |
My notes
Discussion
