version of this book
In this and the following axioms on iteration I introduce structures to perform iteration up to an integer value (a standard loop), a recursive ordinal and larger ordinals defined by a well founded functional hierarchy. In contrast to ordinal iteration in set theory we are always dealing with a process that produces a result in a finite number of operations. The output may be a functional and its range can represent an infinite set. This is not that different than set theory. All sets constructible in a formal system are constructible in a finite number of operations in terms of applying the axioms of the theory to prove the existence of the set. We do not and cannot construct the sets in the domain of the recursive functionals we define but we can always construct the functional itself.