Trustworthiness verification and integrity testing have been identified as key challenges for the sixth generation (6G) of mobile networks and its variety of envisioned features. In this paper, these issues are addressed from a fundamental, algorithmic point of view. For this purpose, the concept of Turing machines is used which provides the fundamental performance limits of digital computers. It is shown that, in general, trustworthiness and integrity cannot be verified by Turing machines and therewith by today's digital computers. In addition, the trustworthiness problem is further shown to be non-Banach-Mazur computable which is the weakest form of computability. Neuromorphic computing has an enormous potential to overcome the limitations of today's digital hardware and, accordingly, it is interesting to study the issues of trustworthiness verification and integrity testing also for such powerful computing models. In particular, as considerable progress in the hardware design for neuromorphic computing has been achieved.