Infinitely large natural numbers, infinitesimally small real numbers, transcendental elements of a field: these are all examples of some sort of "limit" object. In this talk, I'll explain how these concepts are literally limits in the topological sense. We'll introduce lattices, argue that they are the perfect kind of structure for organizing logical information, and then motivate the concept of a filter. Filters show up in both lattice theory and topology, hinting at the connection we will make precise.