Soundness of an arithmetic theory

An arithmetic theory is sound if all its theorems hold in the standard natural numbers. Arithmetic soundness implies consistency and rules out a proof of the negation of a true sentence. It is a semantic property, distinct from effective axiomatizability.

A first-order theory is omega-consistent when it never proves $\exists x\varphi (x)$ while proving $\neg \varphi (\bar{n})$ for every standard natural number $n$. Restricting the matrix to arithmetic bounded arithmetic formulas gives one-consistency. This condition concerns standard instances and is stronger than first-order consistency.

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.

A first-order theory is Sigma-0 sound when every arithmetic bounded arithmetic sentence it proves is true in the standard natural numbers. For a first-order theory proving every true bounded arithmetic sentence, this is equivalent to first-order consistency: a false bounded theorem would have a provable bounded negation, and an inconsistent first-order theory proves the false bounded arithmetic sentence $0=1$.

Pi-1 soundness means that every closed Pi-1 theorem is true in the standard natural numbers. Consistency plus bounded completeness implies it, since a false universal bounded statement has a standard counterexample whose arithmetic verification is provable. It is weaker than full soundness and does not prevent a consistent arithmetic theory from proving a false Sigma-1 sentence.

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.

An arithmetic theory is one-consistent if it does not prove $\exists x\delta (x)$ while proving every $\neg \delta (\overline{n})$, for a bounded arithmetic formula $\delta$ and all standard natural numbers $n$. If the theory proves all true bounded sentences, this is equivalent to Sigma-1 soundness: a false existential theorem has every standard instance refuted by bounded completeness; conversely a true witness and its provable negation contradict the assumption of a consistent first-order theory.

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.