On Tameness, Measurability and the Independence Property

2025/06/10 by Krapp, Lothar Sebastian, Vermeil, Matthieu, Wirth, Laura
#03C40 #03C45 #12J15 #28A05 #54H05 #FOS: Mathematics #Logic (math.LO) #Primary 03C64 #Secondary 12L12

paper · doi:10.48550/arxiv.2506.08733

Abstract

In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield K of the real numbers that lacks several of such tameness properties. As our main result, we present a first-order formula in the language of rings that defines a non-Borel set in K. Moreover, K has the independence property and admits both archimedean and non-archimedean orderings.

Citations

• Measurability in the Fundamental Theorem of Statistical Learning
• Definable Valuations induced by multiplicative subgroups and NIP Fields
• A Guide to NIP theories
• Definability and decision problems in arithmetic

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• Remarks on expansions of the real field: tameness, Hardy fields and smooth rings
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