A proof of completeness for continuous first-order logic

Abstract

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 arbitrary 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.

Effectiveness in RPL, with applications to continuous logic

“…We are grateful to the anonymous referee for the useful comments and critique of the manuscript, and for bringing [BP] to our attention, in which Ben Yaacov and Pedersen present a direct proof of completeness theorem for CL and obtain as a corollary [BP, 9.11] that any computably axiomatizable linear theory is decidable. Unlike [BP], we used an indirect approach of taking a CL theory as a theory in RPL∀ with some additional axioms and using Hajek's result about completeness of RPL∀ to reduce CL to RPL∀ §2.3 and obtain 3.4.…”

Metric structures and probabilistic computation

“…Continuous first-order logic is an extension of Lukasiewicz propositional logic. The following definitions are from [2].…”

“…those accessible to computation), we will not use the stronger notion of a continuous L-structure common in the literature, which requires that ρ be assigned to a complete metric. However, it is possible, given a continuous weak structure (even a pre-structure), to pass to a completion [2]. Definition 2.4.…”

“…The aim of the present section is to prove a similar result for continuous first-order logic and probabilistic computation. The following analogue to the classical concept of the decidability of a theory was proposed in [2]. (1) We define Proof.…”

“…Section 2 will describe the syntax and semantics for continuous first-order logic. The reader familiar with [2] or [1] will find nothing new in Section 2, except a choice of a finite set of logical connectives (no such choice is yet canonical in continuous first-order logic). Section 3 will define probabilistic Turing machines and the class of structures they compute.…”