Theorem 2.2.16 (Crude Incompleteness). Assuming that:

$T$ be a recursive set of (Gödel numberings of) $L_{PA}$ -sentences
$T$ is consistent (never includes both $φ$ and $\neg φ$ )
$T$ contains all the $Σ_{1}$ and $Π_{1}$ sentences provable in PA $^{-}$

Then there is a

Π_{1}

-sentence

τ

such that

τ \notin T

and

\neg τ \notin T

.