Hausdorff dimension, analytic sets and transcendence. (English) Zbl 1082.28002

Summary: Every analytic real closed proper sub-field of \(\mathbb{R}\) has Hausdorff dimension zero. Equivalently, every analytic set of real numbers having positive Hausdorff dimension contains a transcendence base for \(\mathbb{R}\).


28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E15 Descriptive set theory
12L12 Model theory of fields
28A78 Hausdorff and packing measures