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A recursive functional hierarchy

The notation for an integer must, like everything else in the hierarchy, be an instance of the class Functional. We get the integer value from the notation by calling a member function, int_value  . If a Functional is not a notation for an integer this member returns -1.

Class Functional represents arbitrary structures. Integers are treated as a special case with member function int_value. Integers are the finite ordinals. For infinite ordinals we need more structure and this is best handled with a derived class Ordinal  . To determine if an arbitrary functional is an ordinal we will have a member function ordinal_value  . This will return a null pointer if the Functional does not represent an ordinal. If it does represent an ordinal it returns a pointer to itself that can then access member functions specific to ordinals. Functionals that represent integers must be derived from an instance of class Ordinal  and return a pointer from ordinal_value  .

Ordinals must be recursively well ordered. There is a member function order  of Ordinal such that for any two ordinal notations a and b a->Ordinal::order(b) returns -1, 0 or 1 if a is less than, equal to or greater than b. Since b may be from an expansion of the notational hierarchy that a knows nothing about the order  function for a may return -b->order(a). To know whether this is needed there is an Ordinal  member function level  that returns an integer value representing the level in an expanded notational hierarchy. The order  function of the ordinal with the largest value of level must be used. The orderings defined in a particular expansion will always have a fixed recursive ordinal as a limit. This limit does not hold for sequences of expansions.

As mentioned before, in contrast with ZF, there are many types of limit ordinals. Limits are characterized by the type of parameter they allow. The lowest level is the integers. The next level is notations for recursive ordinals (which can be defined as those ordinals represented by structures well founded for the integers). The next levels correspond to structures definable by well foundedness on the previous level of object. We can integrate this up to any recursive ordinal. We want to be able to iterate it up to any recursive notation we can define.

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