If a statement is not provable an inconsistency or self-contradiction may or will develop that invalidates the system.
That's not the case.
You are somehow mixing together the first and the second incompleteness theorem and drawing a misleading conclusion. Briefly put:
The first theorem proves that all consistent axiomatic formulations of number theory which include Peano arithmetic (or stronger) include undecidable propositions.
The second theorem proves that no consistent axiomatic system which includes Peano arithmetic (or stronger) can prove its own consistency.
That the axiomatic system may be inconsistent or not has nothing to do with the existence of undecidable propositions in that system. Inconsistency and incompleteness are not related in that way.
One relation that does hold though is the following: any inconsistent axiomatic system is complete, via explosion principle.
