Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

cognition of structure 59with the sequence of 0s and1s assigned to nodes on that branch up to andincluding the given node; so, for example, the branch up to the leftmost fourthgeneration node (after the initial node) is 〈0, 0, 0, 0〉. A single infinite path upthe tree, a branch, represents the infinite sequence of 0s and1s assigned to itsnodes. Each infinite sequence of 0s and1s is the binary expansion of a realnumber in the closed unit interval. So we can use branches to represent realnumbers in the unit interval. This correlation of sequences with real numbersis not injective: some pairs of sequences represent the same real number. Butwe can easily rectify the matter by cutting out redundant branches.¹⁵ Call theresulting set of branches S. Then there is a one–one correlation of S with[0, 1]: a branch is correlated with the real number of which it is the binaryexpansion. S ordered by ‘