Logical Indeterminacy of the von Neumann Cut: A Structural Obstruction in Formal Theories

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.