 The most famous consequence of the bar theorem is the fan theorem, which suffices to prove the aforementioned theorem on uniform continuity, and which will be treated first. Both the fan and the bar theorem allow the intuitionist to use induction along certain well-founded sets of objects called spreads. A spread is the intuitionistic analogue of a set, and captures the idea of infinite objects as ever growing and never finished. A spread is essentially a countably branching tree labelled with natural numbers or other finite objects and containing only infinite paths.