2. Computability on the Cantor Space

Classically, computability is introduced explicitly for functions f :⊆ (Σ*)n → Σ* on the set Σ* of finite words over an arbitrary finite alphabet Σ, for example by means of Turing machines. For computing functions on other sets M such as natural numbers, rational numbers or finite graphs, words are used as “code words” or “names” of elements of M. Under this view a machine transforms words to words without “understanding” the meaning given to them by the user. Equivalently, one can start with computable functions on the natural numbers instead of words and use numbers as “names”. Since the set Σ* of words over a finite alphabet is only countable, this method cannot be applied for introducing computability on uncountable sets M like the set ℝ of real numbers, the set of open subsets of ℝ or the set C[0; 1] of real continuous functions on the interval [0; 1].