Chapter 7 Degrees of models

Publisher Summary The languages considered in this chapter are recursive and the structures have universe ω. Theories, atomic, complete diagrams of structures, etc., are all identified with subsets of ω through Godel numbering. The chapter discusses structures having “α-th jump degree,” where α is a recursive ordinal. The chapter discusses some results on degrees of models of arithmetic, including the result of Harrington for which he introduced the method of “workers,” and results of Solovay characterizing the degrees of models of various completions of Peano arithmetic (PA). The chapter also discusses some results on the existence of recursive models, including a result of Lerman and Schmerl on models of N0-categorical theories. For some results, the proof is sketched, possibly in a special case, or there is a hint about the main ingredients of the proof. For other results, the proof is not described at all.

[1]  C. Ash,et al.  Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees , 1986 .

[2]  Christopher J. Ash.  A Construction for Recursive Linear Orderings , 1991, J. Symb. Log..

[3]  Alistair H. Lachlan,et al.  Models of Arithmetic and Upper Bounds for Arithmetic Sets , 1994, J. Symb. Log..

[4]  Carl G. Jockusch,et al.  Every low Boolean algebra is isomorphic to a recursive one , 1994 .

[5]  Stephan Wehner,et al.  Enumerations, countable structures and Turing degrees , 1998 .

[6]  Julia F. Knight,et al.  Degrees coded in jumps of orderings , 1986, Journal of Symbolic Logic.

[7]  Alistair H. Lachlan,et al.  Two theorems on degrees of models of true arithmetic , 1984, Journal of Symbolic Logic.

[8]  Kelleen R. Hurlburt.  Sufficiency Conditions for Theories with Recursive Models , 1992, Ann. Pure Appl. Log..

[9]  Julia F. Knight,et al.  Models of Arithmetic and Closed Ideals , 1982, J. Symb. Log..

[10]  Julia A. Knight.  Degrees of models with prescribed Scott set , 1987 .

[11]  Robert I. Soare,et al.  Boolean Algebras, Stone Spaces, and the Iterated Turing Jump , 1994, J. Symb. Log..

[12]  Angus Macintyre,et al.  Degrees of recursively saturated models , 1984 .

[13]  Errata to: “Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees” [Trans. Amer. Math. Soc. 298 (1986), no. 2, 497–514; MR0860377 (87j:03060)] , 1988 .

[14]  John J. Thurber,et al.  Recursive and r.e. quotient Boolean algebras , 1994, Arch. Math. Log..

[15]  Julia F. Knight.  Additive Structure in Uncountable Models for a Fixed Completion of P , 1983, J. Symb. Log..

[16]  Julia A. Knight.  Nonarithmetical a 0 -categorical theories with recursive models , 1994 .

[17]  L. Moss.  Ann. pure appl. logic: Erratum to “coalgebraic logic” 96 (1999) 277–317 , 1999 .

[18]  James H. Schmerl,et al.  Theories with recursive models , 1979, Journal of Symbolic Logic.

[19]  Lawrence Feiner,et al.  Hierarchies of Boolean algebras , 1970, Journal of Symbolic Logic.

[20]  R. Soare,et al.  Π⁰₁ classes and degrees of theories , 1972 .

[21]  Julia F. Knight,et al.  Generic Copies of Countable Structures , 1989, Ann. Pure Appl. Log..

[22]  Theodore A. Slaman,et al.  Relatively recursive expansions II , 1992 .

[23]  David Marker.  Degrees of Models of True Arithmetic , 1982 .

[24]  Julia F. Knight.  Requirement Systems , 1995, J. Symb. Log..

[25]  Julia F. Knight,et al.  Pairs of Recursive Structures , 1990, Ann. Pure Appl. Log..

[26]  Linda Jean Richter.  Degrees of Structures , 1981, Journal of Symbolic Logic.

[27]  Richard Watnick.  A Generalization of Tennenbaum's Theorem on Effectively Finite Recursive Linear Orderings , 1984, J. Symb. Log..

[28]  Christopher J. Ash,et al.  Jumps of orderings , 1990 .