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.

Ancestors (9)

  1. Omega-consistency of an arithmetic theory
  2. Soundness of an arithmetic theory
  3. First-order theory
  4. First-order logic
  5. Mathematical logic
  6. Foundations of mathematics
  7. Area of mathematics
  8. Mathematics
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