Computable Functors and Effective Interpretability

Harrison-Trainor, Matthew; Melnikov, Alexander G.; Miller, Russell; Montalbán, Antonio

doi:10.1017/jsl.2016.12

Cited by 41 publications

(43 citation statements)

References 16 publications

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“…It should be noted that the method used so far to show completeness for spectra of various classes of structures (as described in Section 1) will not suffice for real closed fields. This method can be described as the construction of a computable functor with a computable inverse, usually between an arbitrary graph and a member of the desired class; see [14] for details and definitions, and [5] for an alternative version using effective interpretations of one structure in another. However, such transformations preserve many other properties.…”

Section: Further Results and Questionsmentioning

confidence: 99%

Degree spectra of real closed fields

Miller¹,

González²

2018

Arch. Math. Logic

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Several researchers have recently established that for every Turing degree c, the real closed field of all c-computable real numbers has spectrum {d : d ′ ≥ c ′′ }. We investigate the spectra of real closed fields further, focusing first on subfields of the field R0 of computable real numbers, then on archimedean real closed fields more generally, and finally on nonarchimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of R0 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. * This is a pre-print of an article to be published in the Archive for Mathematical Logic. The final authenticated version will be available online at: https://doi.org/10.1007/s00153-018-0638-z.

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Section: Further Results and Questionsmentioning

confidence: 99%

Degree spectra of real closed fields

Miller¹,

González²

2018

Arch. Math. Logic

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“…The result can be used to translate various results on linear orders [Dow98] into theorems on real closed fields, but typically "loosing one jump" in the process. See also [Mela] for similar result and its consequences for ordered abelian groups, and see [HTMMM16] for a racent general framework on effective functors.…”

Section: Introductionmentioning

confidence: 92%

Torsion-free abelian groups with optimal Scott families

Melnikov¹

2018

J. Math. Log.

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We prove that for any computable successor ordinal of the form α = δ + 2k (δ limit and k ∈ ω) there exists computable torsion-free abelian group (TFAG) that is relatively ∆ 0 α -categorical and not ∆ 0 α−1 -categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σ c α infinitary computable formulae up to automorphism (i.e., has a c.e. uniformly Σ c α -Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits.As far as we are concerned, the problem of finding such optimal examples of (relatively) ∆ 0 α -categorical TFAGs for arbitrarily large α was first raised by Goncharov at least 10 years ago, but it has resisted solution (see, e.g., Problem 7.1 in survey [Mel14]). As a byproduct of the proof, we introduce an effective functor that transforms a 0 -computable worthy labeled tree (to be defined) into a computable TFAG. We expect this technical result will find further applications not necessarily related to categoricity questions.Proof of Lemma 5.2. First, we verify the seemingly obvious claim:

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“…The formal definitions can be found in Appendix B. There is a very strong resemblance between punctual universality and Turing computable universality of a class in [33]. The only essential difference is that we use primitive recursive functionals instead of Turing functionals.…”

Section: Problem 66 Investigate Into the Punctual Degrees Of Some Ementioning

confidence: 99%