The text is slightly in error. It is straightforward to construct a program that is guaranteed to locate an inconsistency if one exists: just have it generate all theorems and stop when it finds an inconsistency. The problem is that it doesn’t ever stop if there isn’t an inconsistency.
This is the difference between decidability and semi-decidability. All the systems covered by Gödel’s completeness and incompletness theorems are semi-decidable, but not all are decidable.
The text is slightly in error. It is straightforward to construct a program that is guaranteed to locate an inconsistency if one exists: just have it generate all theorems and stop when it finds an inconsistency. The problem is that it doesn’t ever stop if there isn’t an inconsistency.
This is the difference between decidability and semi-decidability. All the systems covered by Gödel’s completeness and incompletness theorems are semi-decidable, but not all are decidable.