= Complete omega-consistent arithmetic theories are sound
{title2=$S\vdash X\iff\mathbb N\models X$}
A bounded-complete complete omega-consistent <arithmetic> <first-order theory> is correct on every <arithmetic> <first-order sentence>. Completeness and <first-order consistency> make its theoremhood obey the Boolean <truth tables>. <Omega-consistency> ensures that every proved <existential sentence> has a provable standard numeral instance, since otherwise completeness refutes all such instances. <Induction> on <logical formulas> now identifies theoremhood with standard truth.
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