= Bounded completeness of an arithmetic theory
An <first-order theory>[arithmetic theory] is <bounded completeness of an arithmetic theory>[bounded-complete] if it proves every true closed formula with only bounded <quantifiers>. It then proves every true <Sigma-1 formula>[Sigma-1] sentence by choosing a standard witness. If it is also <consistent first-order theory>[consistent], it cannot prove a false <Pi-1 formula>[Pi-1] sentence: a standard bounded counterexample supplies a provable <contradiction> to that universal theorem. This arithmetic condition is distinct from <bounded completeness of an arithmetic theory>[bounded completeness] of a statistical <first-order model>[model] or family.
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