Arsam's Paradox – Towards an Algebra of Indeterminate Expressions

Problem (Arsam’s Paradox): Any indeterminate expression can be normalized to a variant of , where the precise structure depends on the infinity level to which the form belongs. The essential distinction is between pure infinities and indeterminate expressions involving both vanishing and diverging terms . Solution with Arsam’s Calculus – A Theoretical Framework for Algebraic Handling of Indeterminate Forms I introduce Arsam’s Calculus, a framework for solving indeterminate forms algebraically by interpreting them as variants of , where is understood as a hierarchy of levels. Inspired by Hardy (Orders of Infinity, we formalize a "law of levels" (A number) that assigns meaning to expressions such as , , , and others, without reducing them directly to limits. The key distinction is that Arsamic analysis regards indeterminate expressions as independent algebraic objects, not merely as limit expressions. This framework may provide new insights into the understanding of singularities, complex indeterminate forms, and potentially also applications in physics and cosmology, where infinities frequently arise.