Calculations of Zeno's Dichotomy under a Minimum Traversable Distance and Treatment of Paradoxes in the Physical World

The central difficulty of Zeno's dichotomy paradox is not only whether an infinite division can have a finite sum, but also how an ordered process of motion can be completed and how the terminal event belongs to that process. Because limits and convergent series can establish quantitative relations without necessarily explaining the structure of sequential completion, this paper distinguishes the logical consistency of mathematical infinity from the realizability of physical processes and adopts a minimum indivisible traversable distance as a working assumption. First, I construct an illustration model of a robot performing dichotomous motion. On this basis, I propose two complementary criteria for resolving the paradox: one characterizes the quantitative condition for a dichotomous motion to reach its endpoint, and the other characterizes the process condition under which a mover reaches the endpoint by traversing one of the distances in the dichotomous sequence. Subsequently, calculations are performed based on the model steps, analyze cases of equal-bisection, fewer-bisection, and more-bisection, and use the order of magnitude of the Planck length as a comparable and computable limiting scale without treating it as an empirically established discrete ontology of spacetime. The same model is further applied to the Achilles and the Tortoise paradox, supertasks of Thomson's lamp and the π-robot to examine executable steps, bisection points, task completion, and final states under a minimum traversal distance constraint. The analysis shows that when a physical process is constrained by a minimum traversable distance, the infinite operations in the paradoxes can be transformed into a finite process with an explicit termination condition, the two criteria for resolving the dichotomy paradox are satisfied; Achilles catches up with the tortoise; and the supertask is completed with a numerical result. This paper also discusses the treatment of paradoxes in the physical world, proposing an approach to treat paradoxes separately in the realm of mathematical logic and the physical world. By denying the existence of paradoxes in the physical world, one can infer the properties of the physical world in reverse, such as the finiteness of physical quantities. However, mathematical infinity and physical finiteness are not mutually exclusive: mathematical models can provide effective predictions without requiring real objects to possess exactly the same continuous structure ontologically as the model.