Full Idea
It is a lesson of the first incompleteness theorem that consistency does not entail soundness. If we add the negation of the gödel sentence for PA as an extra axiom to PA, the result is consistent. This negation is false, so the theory is unsound.
Clarification
PA is Peano Arithmetic
Gist of Idea
The first incompleteness theorem means that consistency does not entail soundness
Source
Leon Horsten (The Tarskian Turn [2011], 04.3)
Book Reference
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.52