My usual interpretation of this is not that we have strange proofs, but rather to interpret this as a failure of Godel encodings to completely capture the notio

My usual interpretation of this is not that we have strange proofs, but rather to interpret this as a failure of Godel encodings to completely capture the notion of proof. That is the statement Con(T) isn't really a statement about the consistency of T, but rather a statement that is "close enough" to suffice for many, but not all purposes.