Mathematics’ Limitation Modelling Universal Structures: The “Not” Function—Paradox’s Mechanism in Linguistics, Mathematics, and Physics

The 1-D geometric model studies the structure of states universally closed to the discrete delineation of their properties and defined as infinities. The closure mechanism is the logical “not” function attached to the named property, creating a paradoxical relationship between segments. Two correlated fundamental reference frames are identified. In the first framework, the paradox mechanism prohibits the discrete enumeration of the state’s internal structure. In the second, segments share property for the same infinity but are excluded from common membership due to their paradoxical relationship across the boundary that divides them. The geometric model analyzes the role of the “not” function in linguistics, mathematics, physics, and the generic structure of dimensional development across the quantum to classical platforms. Logical formalisms necessarily discount paradoxes as anomalies open to more advanced understanding, worked around by restrictions to logic or ignored as nonsensical. The 1-D geometric model takes an opposing analytical perspective, considering paradox a fundamental mechanism. The geometric proof examines two constructions of the right triangle within the unit circle. Although the two formats are paradoxical, with the second having no rational basis, the cosine squared calculations agree. Two paradoxical frameworks cohabit within a universal state defined by the cosine squared function. The 1-D model identifies the power function’s systemic limit in modelling universal states that inherently contain the paradox mechanism in their segment relationship.