There are three primary concepts in the initial hierarchy we develop. The first is the notion of a well founded recursive functional hierarchy of typed objects. A functional in the hierarchy accepts objects of a given type as input and outputs objects of a given type. There is a recursive algorithm to determine if an object in the hierarchy is an allowed input. If we keep applying valid inputs to any member of the hierarchy we will in a finite number of steps get an output which is notation for an integer and not a functional.

The next concept is iterating up to a given ordering. This is far more involved in a recursive functional hierarchy than induction up to an ordinal is in ZF. In set theory one only needs to define what one does at 0, successor and limit ordinals to perform induction. In the recursive functional hierarchy there is a hierarchy or limit types for ordinals. The type of limit ordinal can be that of the integers, notations for recursive ordinals, ordinals that are definable by functionals well founded for notations for recursive ordinals, etc.

The third concept is that of using any ordering relationship we define to iterate the notion of well founded hierarchy up to that `ordinal'.

The set theory approach to extending such a hierarchy is to
posit the existence of structures that are inaccessible through
certain kinds of operations. While we can do this it is a weak way
to extend this hierarchy. A stronger way to extend the hierarchy
involves new *kinds* of abstraction and iteration on those
structures. The nature of this abstraction cannot be understood
without working out the details of the recursive functional
hierarchy. It is precisely these details that are washed away with
the seeming more powerful but actually much weaker approach of
conventional set theory.

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