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

Morozov, Andrei and Korovina, Margarita (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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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
Uncontrolled Keywords: CICADA,Σ-definable set, real numbers, computably enumerable open set
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations
MSC 2010, the AMS's Mathematics Subject Classification > 68 Computer science
Depositing User: Dr Margarita Korovina
Date Deposited: 08 Jan 2010
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1385

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