   The idea of the ramified hierarchy has been extremely important in mathematical logic. Russell considered only the finite iteration of the hierarchy: first-order, second-order, etc., but from the beginning, the possibility of extending the ramification transfinitely was considered. Poincaré (1909) mentions the work of Koenig in this direction. For the example above of numbers of different order, he also defines a number to be inductive of order \(\omega\) if it is inductive of all finite orders. He then points out that x+y is inductive of order \(\omega\) if both \(x\) and \(y\) are. This shows that the introduction of transfinite orders can in some case play the role of the axiom of reducibility. Such transfinite extension of the ramified hierarchy was analysed further by Gödel who noticed the key fact that the following form of the reducibility axiom is actually provable: when one extends the ramified hierarchy of properties over the natural numbers into the transfinite this hierarchy collapses at level \(\omega_1\), the least uncountable ordinal (Gödel 1995, and Prawitz 1970). Furthermore, while at all levels \(\lt \omega_1\), the collection of predicates is countable, the collection of predicates at level \(\omega_1\) is of cardinality \(\omega_1\). This fact was a strong motivation behind Gödel’s model of constructible sets. In this model the collection of all subsets of the set of natural numbers (represented by predicates) is of cardinality \(\omega_1\) and is similar to the ramified hierarchy. This model satisfies in this way the Continuum Hypothesis, and gives a relative consistency proof of this axiom. (The motivation of Gödel was originally only to build a model where the collection of all subsets of natural numbers is well-ordered.)