Proof of the Nonexistence of an Algorithm That Tells If a Point Is on a Closed Sets Boundary or Exterior

This study investigates the connection between the undecidability of certain sets and the problem of determining the position of a point relative to the boundary or exterior of a closed set. We introduce a new algorithm, referred to as Algorithm A, which generates a closed set at regular intervals. By employing a reductio ad absurdum argument, we demonstrate that no algorithm exists that can consistently determine whether a point lies on the boundary or in the exterior of a closed set. This finding highlights the limitations of algorithmic approaches in the context of constructive mathematics.