A computability theoretic equivalent to Vaught’s conjecture

We prove that, for every theory T which is given by an Lω1,ω sentence, T has less than 2ℵ0 many countable models if and only if we have that, for every X∈2ω on a cone of Turing degrees, every X-hyperarithmetic model of T has an X-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught’s conjecture.

[1]  Antonio Montalbán,et al.  Boolean algebra approximations , 2014 .

[2]  John R. Steel,et al.  A proof of projective determinacy , 1989 .

[3]  Gerald E. Sacks,et al.  On the Number of Countable Models , 1983 .

[4]  Julia A. Knight,et al.  Computable structures and the hyperarithmetical hierarchy , 2000 .

[5]  Antonio Montalbán.  Counting the back-and-forth types , 2012, J. Log. Comput..

[6]  Antonio Montalbán,et al.  On the Equimorphism Types of Linear Orderings , 2007, Bulletin of Symbolic Logic.

[7]  Richard Laver,et al.  On Fraisse's order type conjecture , 1971 .

[8]  Antonio Montalbán,et al.  Up to equimorphism, hyperarithmetic is recursive , 2005, Journal of Symbolic Logic.

[9]  Theodore A. Slaman,et al.  Definable functions on degrees , 1988 .

[10]  Enrique Casanovas March.  The number of countable models , 2012 .

[11]  Gerald E. Sacks,et al.  Bounds on Weak Scattering , 2007, Notre Dame J. Formal Log..

[12]  Joseph Harrison,et al.  Recursive pseudo-well-orderings , 1968 .

[13]  J. Silver,et al.  Counting the number of equivalence classes of Borel and coanalytic equivalence relations , 1980 .

[14]  Jon Barwise,et al.  Admissible sets and structures , 1975 .

[15]  John R. Steel,et al.  On Vaught's conjecture , 1978 .

[16]  G. Sacks.  Higher recursion theory , 1990 .

[17]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[18]  R. Vaught,et al.  Denumerable Models of Complete Theories , 1970 .

[19]  Antonio Montalbán,et al.  ON THE n-BACK-AND-FORTH TYPES OF BOOLEAN ALGEBRAS , 2012 .

[20]  M. Nadel,et al.  Scott sentences and admissible sets , 1974 .

[21]  Jon Barwise,et al.  Infinitary properties of abelian torsion groups , 1970 .

[22]  Clifford Spector,et al.  Recursive well-orderings , 1955, Journal of Symbolic Logic.

[23]  Antonio Montalbán,et al.  RANKED STRUCTURES AND ARITHMETIC TRANSFINITE RECURSION , 2008 .

[24]  Dana Scott,et al.  LOGIC WITH DENUMERABLY LONG FORMULAS AND FINITE STRINGS OF QUANTIFIERS , 2014 .