Delta Sets as Formal Boundary Objects in Incomplete Theories

This paper introduces Delta sets ($\Delta$) as formally defined regions of non-derivability within a formal theory, establishing a precise link between logical incompleteness and the boundary-based formulations of the TNA framework. We prove that for any incomplete theory, the Delta set is non-empty and constitutes the structural boundary of the system ($\partial\Omega_T = \Delta$). The study demonstrates that these sets induce a partition of the domain and correspond to regions of zero derivability reliability ($R=0$). Furthermore, we provide a formal construction of boundary pressure ($F_B$) derived from the magnitude and weighting of the Delta set, showing that incompleteness fundamentally generates a structural pressure for theoretical extension. This results in a formal grounding for the transition from axiomatic systems to dynamic boundary structures