 In light of the paradoxes for early set theories (Russell’s paradox, the Burali-Forti paradox, and others), some mathematicians and philosophers worried that standard set theory might be inconsistent as well. One alternative viewpoint on mathematics is intuitionism, which only accepts the existence of mathematical objects whose construction can be carried out in some sense by the human mind. Intuitionism requires a revision of logic, since this limitation invalidates the Law of Excluded Middle—there are cases in which we can prove that the non-existence of a certain type of object would lead to a contradiction, but don’t have any method of constructing such an object, so that there might be a truth-value gap. Intuitionists often accept the Aristotelian limitation to “potential infinities”, rather than “actual infinities”, but there is also sophisticated intuitionistic reasoning about what types of infinite entities might exist. (For more, see the entries on  intuitionistic logic  and  intuitionism in the philosophy of mathematics.)