Persistently finite theories with hyperarithmetic models
Author: Terrence Millar
Journal: Trans. Amer. Math. Soc. 278 (1983), 91-99
MSC: Primary 03C57; Secondary 03C50
MathSciNet review: 697062
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Abstract: Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in $0’$. The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.
- C. C. Chang and H. J. Keisler, Model theory, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Studies in Logic and the Foundations of Mathematics, Vol. 73. MR 0409165
- Terrence S. Millar, Foundations of recursive model theory, Ann. Math. Logic 13 (1978), no. 1, 45–72. MR 482430, DOI https://doi.org/10.1016/0003-4843%2878%2990030-X
- T. S. Millar, Type structure complexity and decidability, Trans. Amer. Math. Soc. 271 (1982), no. 1, 73–81. MR 648078, DOI https://doi.org/10.1090/S0002-9947-1982-0648078-8
- Hartley Rogers Jr., Theory of recursive functions and effective computability, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0224462
C. C. Chang and H. J. Keisler, Model theory, American Elsevier, New York, 1973.
T. S. Millar, Foundations of recursive model theory, Ann. Math. Logic 13 (1978), 45-72.
---, Type structure complexity and decidability, Trans. Amer. Math. Soc. 271 (1982), 73-81.
H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.