 According to him, this is not a restriction since—he claims—there are nondenumerably many computable reals in his sense. He also states that it is possible to display the new computable continuum in a hierarchy (i.e., a \(\subset\)-increasing sequence) \(\{B_{\alpha} \mid \alpha \lt \aleph_1\}\) of subsets of \(N^N\), which is defined in such a way that each \(B_{\alpha}\) is at most of cardinality \(\aleph_1\) and hence that the cardinality of the union of the sequence is at most \(\aleph_1\). The idea is to start with a basic domain \(B_0 \subset N^N\) of simple functions (e.g., those having finitely many values), then to define a new domain \(B_1\) which extends \(B_0\) with operations that are defined from elements of \(B_0\), and so on.