This paper establishes the logical indeterminacy of the von Neumann cut within a purely formal setting. The cut is treated as an internal distinction in a first-order axiomatic theory, without recourse to physical, probabilistic, or interpretative assumptions. We introduce an axiomatic framework Fcut , formulated in classical first-order logic and encompassing abstract C* -algebra together with an irreducible update operation. We show that any such framework admitting the iteration and comparison of update outcomes necessarily interprets Robinson arithmetic. As a consequence, Fcut admits Gödel arithmetisation and is therefore essentially incomplete. The indeterminacy of the cut thus arises as a structural property of the formal language itself, and not from any diagonal construction taken as a primitive assumption. This establishes that no consistent extension of Fcut can fix the cut, placing this indeterminacy alongside classical incompleteness phenomena as an intrinsic limitation of sufficiently expressive formal systems.
Emmanuel A. P. Brandt (Mon,) studied this question.