The Impossibility of an Actually Infinite Future: Grounding, Traversal, and Temporal Structure

Arguments against an actually infinite past have traditionally appealed to the impossibility of traversing an actual infinite in order to reach the present. This paper develops a parallel argument against an actually infinite future by introducing a backward-traversal problem: if the future were actually infinite, the present could neither be coherently reached nor fully located within a completed temporal structure. An actually infinite future blocks the present from being fully determined or grounded because locating the present within such a structure would require backward traversal through an unbounded chain of future temporal relations, which cannot be completed in principle. The paper therefore examines whether an actually infinite future is compatible with coherent temporal ordering, locatability, and grounded determination.

The analysis begins from the foundational principle that the future is impossible to be infinite. From this principle, two consequences are distinguished: (1) the future cannot become actually infinite, and (2) the future is finite (if an infinite future is impossible, then a finite future follows directly, since finitude is the opposite of infinitude). The paper argues that these are not independent premises but context-sensitive consequences depending on how the future is understood. If the future is conceived as not yet existing and continuously growing, the first consequence appears to allow a potentially infinite future, since the future never becomes complete. However, the second consequence requires a determinate endpoint and therefore excludes potential infinity. The resulting tension motivates reconsideration of the assumption that the future is merely a growing process. The paper argues that the contradiction is resolved only if the future is understood as already existing and finite.

A central focus of the paper is the objection that locatability does not require traversal. According to relational and eternalist models, moments can allegedly be fully determined by their relations alone, just as elements in infinite mathematical structures are locatable without traversal from infinity. The paper challenges this analogy by distinguishing abstract mathematical ordering from temporal ordering. Mathematical structures are static and non-historical, whereas temporal structures involve asymmetric dependence relations constituting a directed history in which later events are intelligible only through prior events. Because temporal events depend on prior explanatory conditions, a completed temporal structure requires a terminating or well-founded basis for determining the position of events within the whole.

On this basis, the paper reformulates traversal as a grounding condition rather than literal movement through time. Temporal positions must be grounded within a determinate temporal order, and this requires a terminating or well-founded structure of dependence relations. An actually infinite future fails to satisfy this condition because it generates a non-terminating regress of temporal dependence relations extending endlessly beyond the present, preventing complete grounding of temporal positions within the total structure.

The paper concludes that an actually infinite future is incompatible with coherent temporal structure, grounded locatability, and explanatory determination. Because temporal order is historical and dependence-based rather than purely abstract like mathematical order, the future must be finite within any coherent and fully grounded temporal framework.