On Σ‐definability without equality over the real numbers
Andrey S. Morozov, Margarita Korovina · Mathematical logic quarterly · 2008
Abstract In [5] (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 Σ‐definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of Σ‐definable sets (i. e., Σ‐formulas) into new definitions of Σ‐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 Σ‐definability with equality and Σ‐definability without equality. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)