Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes

Details Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes

2010-08-16

arxiv.org/abs/1008.2575v1

Mathematics

Logic

Scientific paper

Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of function composition, division by variables, and extraction of implicitly defined functions. It is shown that if the family $S$ is generic (which is a certain technically defined transcendence condition), then the theory of $\RR_S$ is decidable if and only if $S$ is computably $C^\infty$ (which means that all the partial derivatives of the functions in $S$ may be effectively approximated). It is also shown that, in a certain topological sense, many generic, computably $C^\infty$ families $S$ exist.

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