Model completeness and relative decidability
Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must...
Ausführliche Beschreibung
Gespeichert in:
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Chubb, Jennifer [verfasserIn] |
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Artikel |
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Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Archive for mathematical logic - Springer Berlin Heidelberg, 1988, 60(2021), 6 vom: 03. Jan., Seite 721-735 |
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| Übergeordnetes Werk: |
volume:60 ; year:2021 ; number:6 ; day:03 ; month:01 ; pages:721-735 |
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| DOI / URN: |
10.1007/s00153-020-00753-4 |
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| 520 | |a Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding $$E(\mathcal {A})$$ from the atomic diagram $$\varDelta (\mathcal {A})$$ for all countable models $$\mathcal {A}$$ of T. Moreover, if every presentation of a single isomorphism type $$\mathcal {A}$$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants. | ||
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10.1007/s00153-020-00753-4 doi (DE-627)OLC2126538478 (DE-He213)s00153-020-00753-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Chubb, Jennifer verfasserin (orcid)0000-0001-5754-6563 aut Model completeness and relative decidability 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding $$E(\mathcal {A})$$ from the atomic diagram $$\varDelta (\mathcal {A})$$ for all countable models $$\mathcal {A}$$ of T. Moreover, if every presentation of a single isomorphism type $$\mathcal {A}$$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants. Computability Computable model theory Model completeness Model theory Relative decidability Miller, Russell (orcid)0000-0001-5454-6736 aut Solomon, Reed (orcid)0000-0001-7574-204X aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 60(2021), 6 vom: 03. Jan., Seite 721-735 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:60 year:2021 number:6 day:03 month:01 pages:721-735 https://doi.org/10.1007/s00153-020-00753-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2018 GBV_ILN_4277 AR 60 2021 6 03 01 721-735 |
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Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding $$E(\mathcal {A})$$ from the atomic diagram $$\varDelta (\mathcal {A})$$ for all countable models $$\mathcal {A}$$ of T. Moreover, if every presentation of a single isomorphism type $$\mathcal {A}$$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding $$E(\mathcal {A})$$ from the atomic diagram $$\varDelta (\mathcal {A})$$ for all countable models $$\mathcal {A}$$ of T. Moreover, if every presentation of a single isomorphism type $$\mathcal {A}$$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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Abstract We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $$\mathcal {A}$$ of a computably enumerable, model complete theory, the entire elementary diagram $$E(\mathcal {A})$$ must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure that succeeds in deciding $$E(\mathcal {A})$$ from the atomic diagram $$\varDelta (\mathcal {A})$$ for all countable models $$\mathcal {A}$$ of T. Moreover, if every presentation of a single isomorphism type $$\mathcal {A}$$ has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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Model completeness and relative decidability |
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