# A proof of completeness for continuous first-order logic

Journal of Symbolic Logic 75 (1):168-190 (2010)

# 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.

## PhilArchive

Upload a copy of this work     Papers currently archived: 86,605

Setup an account with your affiliations in order to access resources via your University's proxy server

# Similar books and articles

Continuous first order logic for unbounded metric structures.Itaï Ben Yaacov - 2008 - Journal of Mathematical Logic 8 (2):197-223.
Introduction to mathematical logic.Micha? Walicki - 2012 - Hackensack, NJ: World Scientific.
Russell's completeness proof.Peter Milne - 2008 - History and Philosophy of Logic 29 (1):31-62.
Undecidability and intuitionistic incompleteness.D. C. McCarty - 1996 - Journal of Philosophical Logic 25 (5):559 - 565.
Stability and stable groups in continuous logic.Itaï Ben Yaacov - 2010 - Journal of Symbolic Logic 75 (3):1111-1136.
Intuitionistic completeness for first order classical logic.Stefano Berardi - 1999 - Journal of Symbolic Logic 64 (1):304-312.

2010-09-12

51 (#264,717)

6 months
3 (#344,247)

# Author's Profile

Arthur Paul Pedersen
Carnegie Mellon University (PhD)

# Citations of this work

Omitting types in logic of metric structures.Ilijas Farah & Menachem Magidor - 2018 - Journal of Mathematical Logic 18 (2):1850006.
A completeness theorem for continuous predicate modal logic.Stefano Baratella - 2019 - Archive for Mathematical Logic 58 (1-2):183-201.
The eal 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.

# References found in this work

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.
On Fuzzy Logic III. Semantical completeness of some many-valued propositional calculi.Jan Pavelka - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (25-29):447-464.