Ordinal notation

Abstract: Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1 in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1 itself is assigned a notation.1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (or n-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.