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The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum

B. L. Budinas


Abstract: A set is called a selector of an equivalence relation defined on all the real numbers if it intersects each equivalence class of this relation in a singleton set. The following proposition is called the selector principle: each analytic equivalence relation on the set of all real numbers has an $A_2$-selector. It is proved that the selector principle is not equivalent to the existence of an $A_2$ well ordering of the continuum. This answers a question posed by Burgess. Equivalence is understood in the sense of equivalence in the standard Zermelo–Fraenkel set theory with the axiom of choice.
Bibliography: 8 titles.

UDC: 519.5

MSC: Primary 04A15, 04A99, 06A99, 54C65; Secondary 03E40, 03F65, 28A05

Received: 29.12.1980

 English version: Mathematics of the USSR-Sbornik, 1984, 48:1, 159–172

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