 From a classical viewpoint, these results fit into place when one realises that words such as “function” and “real number” should be interpreted as “recursive function” and “recursive real number” respectively. Note that the second of the above four recursive theorems is a strong recursive counterexample to the open-cover compactness property of the (recursive) real line; and the fourth is a recursive counterexample to the classical theorem that every uniformly continuous mapping of a compact set into \(\bR\) attains its infimum.