What is o-minimality?
Let \(\mathbb R\) denote the set of real numbers and let \(W\) be a set of functions, each from some \(\mathbb R^n\) to some \(\mathbb R^m\). Friedman proves that \((\mathbb R,<, 0,1,+,-,\times, W)\) is o-minimal if and only if there exists a least \(V\) such that \(W\subseteq V\) and: \(V\) contains all polynomial functions; it is is closed under compositions; for all \(f,g\) in \(V\), if \(f\) and \(g\) are of the same arity, then \(h(x)=(f(x),g(x))\) is in \(V\); if \(f:\mathbb R^{n+m}\to \mathbb R\) is in \(V\) then there is a \(g\) in \(V\) such that for all \(x\), if \(f(x,y)=0\) for some \(y\), then \(f(x,g(x))=0\); if \(f:\mathbb R\to \mathbb R\) in \(V\) is bounded, then \(f\) has a limit at infinity. The result is then generalized to arbitrary ordered rings \(R\), but to this end the last condition above has to be replaced by: for all \(f: R\to R\) in \(V\) there exist \(x_1<\dots <x_k\in R\), \(k\geq 1\), such that on each of the intervals \((-\infty, x_1), (x_1,x_2),\dots, (x_k,\infty)\), \(f\) is either constant, strictly increasing, or strictly decreasing.
scientific article; zbMATH DE number 5378326
scientific article