A proof of completeness for continuous first-order logic

TL;DR

This paper proves that continuous first-order logic has a completeness property where satisfiability aligns with consistency, and it establishes an approximate strong completeness theorem allowing finite proofs to approximate truth arbitrarily closely.

Contribution

It provides a proof of completeness for continuous first-order logic, including an approximate strong completeness theorem, which was previously not established.

Findings

Satisfiability in continuous first-order logic is equivalent to consistency.

Continuous first-order logic satisfies an approximate form of strong completeness.

Finite proofs can approximate the truth of formulas arbitrarily closely.

Abstract

The primary purpose of this article is to show that a certain natural set of axioms yields 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 \emph{approximated} form of strong completeness, whereby $\Sigma\vDash\varphi$ (if and) only if $\Sigma\vdash\varphi\dotminus 2^{-n}$ for all $n<\omega$. This approximated form of strong completeness asserts that if $\Sigma\vDash\varphi$, then proofs from $\Sigma$, being finite, can provide arbitrary better approximations of the truth of $\varphi$.