What does Godels Incompleteness theorem "true but unprovable" mean?
Your difficulties are hinting toward Gödel's second incompleteness theorem. The proof system $\Pi$ can formalize most steps in the proof, other than its own consistency – the fact that $\Pi$ cannot prove a statement and its negation. Following exactly this train of though, Gödel showed that a sufficiently strong proof system cannot prove its own consistency.

For more information, I suggest looking up resources on Gödel's second incompleteness theorem.