Real Analysis Exchange

No transcendence basis of R over Q can be analytic

Enrico Zoli

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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 $\mathbb{R}$.

Article information

Real Anal. Exchange, Volume 30, Number 1 (2004), 311-318.

First available in Project Euclid: 27 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 12F20: Transcendental extensions 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

algebraically independent sets analytic sets Borel classes


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

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