Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

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Title: Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ? - MathOverflow
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Answer by Joel David Hamkins for Is the equivalence between a $\Sigma^0_1$...

May 19, 2013, 3:20 am

No, in general, a true $\Delta^0_1$ assertion may not necessarily be provably $\Delta^0_1$ in a given theory. For example, assume $\text{Con}(\text{PA})$ is true, and consider the formula $\phi(a)$...

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Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining...

July 4, 2013, 11:08 pm

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no...

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