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arXiv · 0903.4051

A proof of completeness for continuous first-order logic

Itaï Ben Yaacov·Arthur Paul Pedersen
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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 $Σ\vDashφ$ (if and) only if $Σ\vdashφ\dotminus 2^{-n}$ for all $n<ω$. This approximated form of strong completeness asserts that if $Σ\vDashφ$, then proofs from $Σ$, being finite, can provide arbitrary better approximations of the truth of $φ$.

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Itaï Ben Yaacov, Arthur Paul Pedersen. 2009-03-24. A proof of completeness for continuous first-order logic. https://doi.org/10.2178/jsl/1264433914

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