David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
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 φT 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 φT∘ is uniformly recursive from φ, where φT∘ 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
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Itaï Ben Yaacov & Arthur Paul Pedersen (2010). A Proof of Completeness for Continuous First-Order Logic. Journal of Symbolic Logic 75 (1):168-190.
Richard Kaye (2007). The Mathematics of Logic: A Guide to Completeness Theorems and Their Applications. Cambridge University Press.
Alexander Paseau (2010). Pure Second-Order Logic with Second-Order Identity. Notre Dame Journal of Formal Logic 51 (3):351-360.
Marcus Rossberg (2004). First-Order Logic, Second-Order Logic, and Completeness. In Vincent Hendricks, Fabian Neuhaus, Stig Andur Pedersen, Uwe Scheffler & Heinrich Wansing (eds.), First-Order Logic Revisited. Logos. 303-321.
D. C. McCarty (1996). Undecidability and Intuitionistic Incompleteness. Journal of Philosophical Logic 25 (5):559 - 565.
Stephen Read (1997). Completeness and Categoricity: Frege, Gödel and Model Theory. History and Philosophy of Logic 18 (2):79-93.
Itaï Ben Yaacov (2008). Topometric Spaces and Perturbations of Metric Structures. Logic and Analysis 1 (3-4):235-272.
Michał Walicki (2012). Introduction to Mathematical Logic. World Scientific.
Henri Galinon (2009). A Note on Generalized Functional Completeness in the Realm of Elementrary Logic. Bulletin of the Section of Logic 38 (1):1-9.
Silvio Ghilardi & Pierangelo Miglioli (1999). On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics. Studia Logica 63 (3):353-385.
Stefano Berardi (1999). Intuitionistic Completeness for First Order Classical Logic. Journal of Symbolic Logic 64 (1):304-312.
Jouko Väänänen (2012). Second Order Logic or Set Theory? Bulletin of Symbolic Logic 18 (1):91-121.
Enrico Martino (1998). Negationless Intuitionism. Journal of Philosophical Logic 27 (2):165-177.
Vilém Novák (1987). First-Order Fuzzy Logic. Studia Logica 46 (1):87 - 109.
Douglas Cenzer & Jeffrey B. Remmel (2006). Complexity, Decidability and Completeness. Journal of Symbolic Logic 71 (2):399 - 424.
Added to index2010-09-14
Total downloads18 ( #107,505 of 1,684,545 )
Recent downloads (6 months)2 ( #112,061 of 1,684,545 )
How can I increase my downloads?