idea
Gödel’s second theorem
The reasoning “if the system is consistent, the sentence can’t be proven” can itself be written down inside the system. So the system can prove: “if I’m consistent, then the Gödel sentence is true”.
Now suppose it could also prove “I am consistent”. It could combine the two and prove the Gödel sentence. But we just saw it cannot, unless it is broken. So it cannot prove its own consistency either.