Journal of Symbolic Logic 75 (1):168-190 (2010)
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Abstract |
Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?
The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.
Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.
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DOI | 10.2178/jsl/1264433914 |
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References found in this work BETA
Positive Model Theory and Compact Abstract Theories.Itay Ben-Yaacov - 2003 - Journal of Mathematical Logic 3 (01):85-118.
Simplicity in Compact Abstract Theories.Itay Ben-Yaacov - 2003 - Journal of Mathematical Logic 3 (02):163-191.
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Citations of this work BETA
Omitting Types in Logic of Metric Structures.Ilijas Farah & Menachem Magidor - 2018 - Journal of Mathematical Logic 18 (2):1850006.
The Real Truth.Stefano Baratella & Domenico Zambella - 2015 - Mathematical Logic Quarterly 61 (1-2):32-44.
A Note on Infinitary Continuous Logic.Stefano Baratella - 2015 - Mathematical Logic Quarterly 61 (6):448-457.
Continuous Propositional Modal Logic.Stefano Baratella - 2018 - Journal of Applied Non-Classical Logics 28 (4):297-312.
A Completeness Theorem for Continuous Predicate Modal Logic.Stefano Baratella - 2019 - Archive for Mathematical Logic 58 (1-2):183-201.
View all 7 citations / Add more citations
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