Low basis theorem
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree formula_1, it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low; that is, the Turing jump of the path is Turing equivalent to the halting problem formula_2. Statement and proof. The low basis theorem states that every nonempty formula_3 class in formula_4 (see arithmetical hierarchy) contains a set of low  degree (Soare 1987:109). This is equivalent, by definition, to the statement that each infinite computable subtree of the binary tree formula_1 has an infinite path of low degree. The proof uses the method of forcing with formula_3 classes (Cooper 2004:330). Hájek and Kučera (1989) showed that the low basis is provable in the formal system of arithmetic known as formula_7. Application. One application of the low basis theorem is to construct completions of effective theories so that the completions have low Turing degree. For example, the low basis theorem implies the existence of PA degrees strictly below formula_2. Low basis