Abstract
We give an example of an iteration with recursive data which stabilises exactly at the first non-recursive ordinal. We characterise the points in the final set as those attacked by recurrent points, and use that characterisation to show that recurrent points must exist for any iteration with recursive data which does not stabilise at a recursive ordinal.
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References
K.J. Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin — Heidelberg — New York, 1975.
K.J. Barwise, R.O.Gandy and Y.N.Moschovakis, The next admissible set, Journal of Symbolic Logic 36 (1971) 108–120.
J. Harrison, Recursive pseudo-well-orderings, Transactions of the American Math-ematical Society 131 (1968) 526–543. For a review, see [10].
F. Hausdorff, Grundzüge der Mengenlehre, W. de Gruyter, Leipzig, 1914. (available in an English translation)
G. Kreisel, Analysis of Cantor-Bendixson theorem by means of the analytic hierarchy, Bulletin de l’Academie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, 7 (1959) 621–626.
R. Mansfield and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, Oxford Logic Guides, #11, Oxford University Press, 1985.
A.R.D. Mathias, An application of descriptive set theory to a problem in dynamics, CRM Preprint num. 308, Octubre 1995 (See also ardm/appdyn97).
A.R.D. Mathias, Long delays in dynamics, CRM Preprint num. 334, Maig 1996 (See also ardm/delays97).
Y.N. Moschovakis, Descriptive Set Theory, North Holland, Amsterdam — New York — Oxford, 1980.
Y.N. Moschovakis, Review of [3], Journal of Symbolic Logic 37 (1972) 197–8.
Y.N. Moschovakis, Review of [5], Journal of Symbolic Logic 35 (1970) 334.
H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
J. Shoenfield, Mathematical Logic, Addison-Wesley, Reading, Massachusetts, 1967.
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© 1998 Springer Science+Business Media Dordrecht
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Mathias, A.R.D. (1998). Recurrent Points and Hyperarithmetic Sets. In: Di Prisco, C.A., Larson, J.A., Bagaria, J., Mathias, A.R.D. (eds) Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8988-8_10
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DOI: https://doi.org/10.1007/978-94-015-8988-8_10
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