Intrinsically hyperarithmetical sets. (English) Zbl 0859.03016
Summary: The main result proved is that on every recursive structure the intrinsically hyperarithmetical sets coincide with the relatively intrinsically hyperarithmetical sets. As a side effect of the proof an effective version of Kueker’s theorem on definability by means of infinitary formulas is obtained.
Keywords:
abstract computability; external definability; formal definability; enumerations; forcing; recursive structure; intrinsically hyperarithmetical sets; Kueker’s theoremReferences:
| [1] | Ash, Ann. Pure Appl. Logic 42 pp 195– (1989) |
| [2] | Ash, Trans. Amer. Math. Soc. 298 pp 497– (1986) |
| [3] | and , Intrinsically recursive relations. In: Aspects of Effective Algebra (ed.), Upside Down A Book Co., Steel’s Creek, Australia, 1981, pp. 26–41. |
| [4] | Chisholm, J. Symbolic Logic 55 pp 1213– (1990) |
| [5] | Chisholm, J. Symbolic Logic 55 pp 1168– (1990) |
| [6] | Grilliot, J. Symbolic Logic 37 pp 81– (1972) |
| [7] | Definability, automorphisms and infinitary languages. In: The Syntax and Semantics of Infinitary Logic. Lecture Notes in Mathematics 72 (ed.), Springer-Verlag, Berlin-Heidelberg-New York 1968, pp. 152–165. |
| [8] | Lacombe, C. R. de l’Academie des Sciences de Paris 258 pp 3410– (1964) |
| [9] | Techniques and counterexamples in almost categorical recursive model theory. Ph. D. Thesis, University of Wisconsin, Madison, Wisconsin, 1982. |
| [10] | Mansfield, Pacific J. Math 35 pp 451– (1970) · Zbl 0251.02060 · doi:10.2140/pjm.1970.35.451 |
| [11] | Moschovakis, Trans. Amer. Math. Soc. 138 pp 427– (1969) |
| [12] | Moschovakis, Trans. Amer. Math. Soc. 138 pp 465– (1969) |
| [13] | Elementary Induction on Abstract Structures. North-Holland Publ. Comp., Amsterdam 1974. · Zbl 0307.02003 |
| [14] | Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York 1967. · Zbl 0183.01401 |
| [15] | Mathematical Logic. Addison-Wesley Publ. Comp., Reading, Mass., 1967. |
| [16] | Soskov, Math. Logic Quart. 41 pp 109– (1995) |
| [17] | Soskov, J. Symbolic Logic 54 pp 428– (1989) |
| [18] | Soskov, Arch. Math. Logic 29 pp 187– (1990) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
