This is not a weird example for that reason. For a given model, every statement is either true or false. Godel's Completeness Theorem says that a first-order theory is consistent if and only if it is true in some model. Therefore, for every undecidable statement there will be a model where it is true and a model where it is false.
These models can look very strange. For example, if ZF is consistent, then by the Second Incompleteness Theorem so is ZF + "ZF is inconsistent". By the Completeness Theorem, a model for this theory must exist. In this model ZF is inconsistent, so there is a 'proof' of a contradiction from the axioms of ZF. However, since we have assumed the consistency of ZF, such a 'proof' must necessarily involve nonstandard integers.