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On the index sets of Σ-subsets of the real numbers

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Abstract

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the Σ-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of ℝn, first category subsets in ℝn as well as the sets of pairs of Σ-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the Σ-formulas defining connected sets is at least Π 11 .

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. S. Morozov

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  1. A. S. Morozov
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Correspondence to A. S. Morozov.

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Original Russian Text Copyright © 2008 Morozov A. S.

The author was supported by the International Russian-German grant DFG-RFBR No. 06-01-04002-NNIOa, the Russian Foundation for Basic Research (Grant 08-01-00336-a), and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-335.2008.1).

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Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1351–1360, November–December, 2008.

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Morozov, A.S. On the index sets of Σ-subsets of the real numbers. Sib Math J 49, 1078–1084 (2008). https://doi.org/10.1007/s11202-008-0103-z

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  • DOI: https://doi.org/10.1007/s11202-008-0103-z

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