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Assuming PA is consistent, the theory $S=\mathrm{PA}+\neg {\mathrm{Con}}_{\mathrm{PA}}$ is consistent by second incompleteness. Its false axiom $G=\neg {\mathrm{Con}}_{\mathrm{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow \neg {\mathrm{Pr}}_S(┌G┐)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical Gödel sentence. A canonical Pi-1 construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.

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

A Sigma-1 formula is a formula equivalent in arithmetic to $\exists y_1\cdots \exists y_r\delta$, where $\delta$ is bounded. A Pi-1 formula is similarly equivalent to $\forall y_1\cdots \forall y_r\delta$ with $\delta$ bounded.

A Sigma-1 formula is a formula equivalent in first-order arithmetic towhere $\delta$ is bounded. A Pi-1 formula is similarly equivalent to $\forall y_1\cdots \forall y_k\delta$ with bounded $\delta$.

A Pi-1 formula in the arithmetical hierarchy has the form $\forall nR(n,\mathbf{x})$, with $R$ a decidable predicate; several universal number quantifiers can be combined by coding tuples. In a formal arithmetic presentation, the matrix is taken to have bounded quantifiers, with equivalent computable-predicate normal forms obtained by coding computations.

For example, the assertion that a fixed Turing machine never halts is $\forall tR(t)$, where $R(t)$ checks that it has not halted in its first $t$ steps. A false such assertion has a finite counterexample, while a true one need not have a finite computational certificate. Negating a Pi-1 formula gives a Sigma-1 formula.

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.