= One-consistency
{title2=$1\text{-consistency}$}
= One-consistent
{synonym}
= One-consistency of an arithmetic theory
{synonym}
An <arithmetic> theory is one-consistent if it does not prove $\exists x\,\delta(x)$ while proving every $\neg\delta(\overline n)$, for a <bounded arithmetic formula> $\delta$ and all standard <natural numbers> $n$. If the theory proves all true bounded sentences, this is equivalent to <Sigma-1 soundness>: a false existential theorem has every standard instance refuted by bounded completeness; conversely a true witness and its provable <negation> contradict the assumption of a <consistent first-order theory>.
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