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On Σ-definability without equality over the real numbers

✍ Scribed by Andrei S. Morozov; Margarita V. Korovina

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
139 KB
Volume
54
Category
Article
ISSN
0044-3050

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✦ Synopsis

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)

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The expression q n(n&1)Â4 should be replaced with the expression q (n+2)(n&1)Â4 in the first displayed equation in the statement of Theorem 1.2 (page 427), as well as in the first displayed equation in the statement of Proposition 3.2.1 (page 445) and in the displayed equation at the bottom of page