James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of “internal set theory” (a version of nonstandard analysis), developed by Edward Nelson, shows Thomson's lamp is chimerical; its copy within set theory yields a contradiction. The demonstration extends to placing restrictions on other “infinite tasks” such as Zeno's paradoxes (...) of motion and Kant's First Antinomy. Resolution of such logical-philosophical problems leads to some very general constraints which must be placed upon the syntax of physical theories. In particular, at some scale space and time would appear granular. The suitability of internal set theory for analyzing phenomena is examined, using a paper by Alper and Bridger (1997) to frame the discussion. (shrink)