Abstract
Several researchers have recently established that for every Turing degree \(\varvec{c}\), the real closed field of all \(\varvec{c}\)-computable real numbers has spectrum \( \{ \varvec{d} : \varvec{d}'\ge \varvec{c}'' \} \). We investigate the spectra of real closed fields further, focusing first on subfields of the field \(\mathbb {R}_{\varvec{0}}\) of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of \(\mathbb {R}_{\varvec{0}}\) with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.
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Russell Miller: The first author was partially supported by Grant # DMS—1362206 from the National Science Foundation and by several grants from the City University of New York PSC-CUNY Research Award Program. The authors wish to acknowledge useful conversations with Julia Knight and Reed Solomon.
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Miller, R., Ocasio González, V. Degree spectra of real closed fields. Arch. Math. Logic 58, 387–411 (2019). https://doi.org/10.1007/s00153-018-0638-z
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DOI: https://doi.org/10.1007/s00153-018-0638-z