## 2010.8: On $\Sigma$-definability without equality over the real numbers

2010.8:
Andrei Morozov and Margarita Korovina
(2008)
*On $\Sigma$-definability without equality over the real numbers.*
Mathematical Logic Quarterly, 54 (5).
pp. 535-544.
ISSN 0942-5616

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## Abstract

In Delzell (1982) it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for $\Sigma$-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of -definable sets (i. e., $\Sigma$-formulas) into new definitions of $\Sigma$-definable sets in such a way that the results will define open sets, and if a definition defines an open set, then the result of this transformation will define the same set. These results highlight the important differences between $\Sigma$-definability with equality and $\Sigma$-definability without equality.

Item Type: | Article |
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Uncontrolled Keywords: | CICADA,Σ-definable set, real numbers, computably enumerable open set |

Subjects: | MSC 2000 > 03 Mathematical logic and foundations MSC 2000 > 68 Computer science |

MIMS number: | 2010.8 |

Deposited By: | Dr Margarita Korovina |

Deposited On: | 08 January 2010 |

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