Relationship To Constructive Mathematics and Compactness
The fan theorem of Brouwer is, from a classical point of view, the contrapositive of a form of König's lemma. A subset S of is called a bar if any function from to the set has some initial segment in S. A bar is detachable if every sequence is either in the bar or not in the bar (this assumption is required because the theorem is ordinarily considered in situations where the law of the excluded middle is not assumed). A bar is uniform if there is some number N so that any function from to has an initial segment in the bar of length no more than . Brouwer's fan theorem says that any detachable bar is uniform.
This can be proven in a classical setting by considering the bar as an open covering of the compact topological space . Each sequence in the bar represents a basic open set of this space, and these basic open sets cover the space by assumption. By compactness, this cover has a finite subcover. The N of the fan theorem can be taken to be the length of the longest sequence whose basic open set is in the finite subcover. This topological proof can be used in classical mathematics to show that the following form of König's lemma holds: for any natural number k, any infinite subtree of the tree has an infinite path.
Read more about this topic: König's Lemma
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