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Complexity of structures associated with real closed fields
Author(s) -
Knight Julia F.,
Lange Karen
Publication year - 2013
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds070
Subject(s) - mathematics , real number , countable set , residue field , field (mathematics) , conjecture , integer (computer science) , section (typography) , discrete mathematics , group (periodic table) , additive group , ordered field , combinatorics , pure mathematics , computer science , chemistry , organic chemistry , programming language , operating system
Real closed fields, and structures associated with them, are interesting from the point of view of both model theory and computability. In this paper, we give results on the complexity of value group sections and residue field sections. It is not difficult to show that for any countable real closed field R , there is a value group section that isΔ 2 0 ( R ) . This result is sharp in the sense that there is a computable real closed field for which every value group section codes the halting set. For a real closed field R , there is a residue field section that isΠ 2 0 ( R ) . This result is sharp in the sense that there is a computable real closed field R with noΣ 2 0value field section. We are also interested in integer parts. Mourgues and Ressayre showed, by a rather complicated construction, that every real closed field has an integer part. The construction becomes canonical once we fix the real closed field R , a residue field section k , and a well ordering of R . The construction involves mapping the elements of R to generalized series, called developments , with terms corresponding to elements of the natural value group and coefficients in k . The complexity of the construction is clearly related to the lengths of the developments. We conjecture that for a type ω well ordering on R , the lengths of the developments are less thanω ω ω. We give an example showing that there is no smaller ordinal bound.