User blog comment:PolyhedralMatrix102/"Bigger" notation/@comment-32783837-20180319182438/@comment-1605058-20180319191646

It actually is possible. The thing, however, is that the theory has to prove some statements which are false. For example, if ZFC is consistent, then ZFC + "ZFC is inconsistent" has PTO exactly equal to the CK ordinal. If we assume it only proves true sentences, however (a property known as soundness), then the PTO has to be strictly smaller, as you say.