2006/06/30 by Luck Darnière, Darnière, Luck
Computer Science · Mathematics · #03C10 #06D20 #06D99 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.math/0606792
openalex publication_date 2006/06/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
It is known from Grzegorczyk's paper \citegrze-1951 that the lattice of real semi-algebraic closed subsets of \mathbb Rn is undecidable for every integer n≥ 2. More generally, if X is any definable set over a real or algebraically closed field K, then the lattice L(X) of all definable subsets of X closed in X is undecidable whenever dim X≥ 2. Nevertheless, we investigate in this paper the model theory of the class \rm SC_def(K,d) of all such lattices L(X) with dim X≤ d and K as above or a henselian valued field of characteristic zero. We show that the universal theory of \rm SC_def(K,d), in a natural expansion by definition of the lattice language, is the same for every such field K. We give a finite axiomatization of it and prove that it is locally finite and admits a model-completion, which turns to be decidable as well as all its completions. We expect L(\mathbb Q_pd) to be a model of (a little variant of) this model-completion. This leads us to a new conjecture in p-adic semi-algebraic geometry which, combined with the results of this paper, would give decidability (via a natural recursive axiomatization) and elimination of quantifiers for the complete theory of L(\mathbb R_pd), uniformly in p.