## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 75, Issue 1 (2010), 168-190.

### A proof of completeness for continuous first-order logic

Arthur Paul Pedersen and Itaï Ben Yaacov

#### 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}^{∘ }is uniformly recursive from φ, where
φ_{T}^{∘}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.

#### Article information

**Source**

J. Symbolic Logic Volume 75, Issue 1 (2010), 168-190.

**Dates**

First available in Project Euclid: 25 January 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1264433914

**Digital Object Identifier**

doi:10.2178/jsl/1264433914

**Mathematical Reviews number (MathSciNet)**

MR2605887

**Zentralblatt MATH identifier**

1194.03028

#### Citation

Yaacov, Itaï Ben; Pedersen, Arthur Paul. A proof of completeness for continuous first-order logic. J. Symbolic Logic 75 (2010), no. 1, 168--190. doi:10.2178/jsl/1264433914. https://projecteuclid.org/euclid.jsl/1264433914