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2004-2005 No transcendence basis of R over Q can be analytic
Enrico Zoli
Real Anal. Exchange 30(1): 311-318 (2004-2005).


It has been proved by Sierpiński that no linear basis of $\mathbb{R}$ over $\mathbb{Q}$ can be an analytic set. Here we show that the same assertion holds by replacing ``linear basis'' with ``transcendence basis''. Furthermore, it is demonstrated that purely transcendental subfields of $\mathbb{R}$ generated by Borel bases of the same cardinality are Borel isomorphic (as fields). Following Mauldin's arguments, we also indicate, for each ordinal $\alpha$ such that $1\leq \alpha\lt\omega_1$ ($2\leq \alpha\lt\omega_1$), the existence of subfields of $\mathbb{R}$ of exactly additive (multiplicative, ambiguous) class $\alpha$ in $\mathbb{R}$.


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Enrico Zoli. "No transcendence basis of R over Q can be analytic." Real Anal. Exchange 30 (1) 311 - 318, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 27 July 2005

zbMATH: 1061.28001
MathSciNet: MR2127535

Primary: 12F20 , 28A05

Keywords: algebraically independent sets , analytic sets , Borel classes

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 1 • 2004-2005
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