= Complete one-consistent theories can prove false sentences
Given a bounded-complete <one-consistent> <arithmetic> <first-order theory> with a false theorem, enumerate <first-order sentences> and successively choose <one-consistent> one-sentence extensions. Every stage has <Sigma-1 soundness>, and a false Sigma-1 theorem in the <union> would already have a <formal proof> at one stage. The <union> is therefore <one-consistent> and complete while retaining the false theorem. It cannot be omega-consistent, since a complete bounded-complete omega-consistent <first-order theory> is <sound>.
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