Monthly Archives: October 2024

In our proof of the Cell Decomposition Theorem, we took as known the Uniform Finiteness Theorem.  In this post, we fill that gap. Uniform Finiteness Theorem:   Let $Y \subseteq R^{n+1}$ be a definable set for which all fibers $Y_a := … Continue reading →

The Cell Decomposition Theorem might be called the “Fundamental Theorem of O-minimality”.   With this theorem we show that from the hypothesis that the definable sets in one variable are particularly simply, i.e. those which admit a quantifier-free definition in the … Continue reading →

A basic principle in tame geometry is that there are no pathological definable functions in o-minimal structures.   One precise sense in which this principle is true is given by the Monotonicity Theorem. Monotonicity Theorem :  Let $f: R \to R$ … Continue reading →