Sigma-1 soundness (${\Sigma }_1\text{-soundness}$)

A theory has Sigma-1 soundness if every Sigma-1 formula that is a closed theorem is true in the standard natural-number structure. It is weaker than full arithmetic soundness. Under bounded completeness it is equivalent to one-consistency. The standard interpretation matters: an existential statement can have a witness in a nonstandard model while having no standard witness.