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[Note to editor. The technical work in this appendix is under development. It is possible this will lead to new results publishable in a mathematics journal. The book does not depend on such results and I can complete an abbreviated version of this appendix in a few days if needed.]
This appendix develops a recursive functional hierarchy as an alternative approach for the foundations of mathematics. Instead of speaking of arbitrary sets we will speak of the range of a recursive functional. This does not limit us to recursive sets because the domain of the recursive functional need not be a recursive set. For example the domain might be notations for recursive ordinals.
It may seem to require conventional set theory to define that domain. However one does not need to define the set of all notations for recursive ordinals to provide a suitable definition. One can define what a recursive ordinal notation is as a recursively defined property. For example we can define a recursive functional as being well founded for the integers as a way of defining the recursive ordinals. To give this definition we set up a notation for integers and TM s such that a symbol in this notation describes either an integer or the Gödel number of a TM that accepts an integer as input and has an element of this notation as output. In set theory we would define a TM as being well founded for the integers if and only if for any infinite sequence of integers we can apply the first integer to the original TM , the second integer to the output of the original TM , the third integer to the output of the output of the original TM , etc, and in a finite number of steps we will get the notation for an integer and not another TM as output.
Without quantifying over the reals we can define this a property of TM s . We will let be true if n denotes an integer and otherwise denote output of the TM encoded by n applied to input x. In the following n is any element of the notation.
This recursive definition does not exclude infinite descending chains but it will only allow us to build up structures that are well founded in the set theory sense. This is somewhat analogous to what happens in conventional ZF. We can define the set of all recursive ordinal notations but there is no general way to determine what elements belong to that set. The difference is that we do not speak of the set as if it were a completed entity. We only speak of properties of TM s.
Of course it is useful to talk about the set of all recursive ordinals and I am not advocating that we abandon thinking or doing mathematics in that way. I am advocating that we think of these sets as defined by properties of TM 's. We should draw a line between the structures in ZF that can be interpreted in this way and those which can only be interpreted as properties of a formal system. The purpose of drawing this line is not to weaken but to strengthen mathematics. Mathematics that goes past this line I call shadow mathematics. It is is not about objective properties of TM s but denotes ways in which we can extend an existing formal system to extend those properties. Recognizing where this distinction lies and focusing on extending mathematics at that critical point is essential to producing strong extensions. Working in shadow mathematics is a bit like iterating the consistency of a system. You can always play that game but it is always a weak generalization compared to what is possible through a deeper understanding of the system. As we shall see, the concept of an uncountable ordinal bypasses the generalizations of the concept of an ordinal that are necessary to extend the concept within a recursive functional hierarchy. Loosely speaking, if a statement refers to a recursively enumerable collection of events, then it is a statement about a recursive process in a potentially infinite universe and is objectively true or false. Statements, such as the continuum hypothesis, that cannot be so interpreted, are not objectively true or false. They are part of shadow mathematics. They can gives insight into good ways to extend a system but they are not themselves valid extensions.
The seemingly weaker approach of actually constructing the recursive functionals at each level in the hierarchy will turn out to be more powerful. When we reach the level in this functional hierarchy where induction up to an ordinal is exhausted we need different generalizations for induction and these will eventually lead us outside of what is definable in ZF in a much more powerful way then large cardinal axioms or other conventional ways of extending ZF. The latter are a way of saying that an object exists that is in some sense inaccessible through certain operations. The functional hierarchy approach does something much more powerful. It allows us to define new kinds of abstractions and new kinds of induction on those abstractions.
All the ordinals in our hierarchy must have a recursive ordering relationship or we cannot not use them in recursive functionals. This implies that there is a recursive ordinal that is the limit of everything we can define. We get around this by making our definitions expandable so we can add new functionals. For every recursive ordinal there must be some expansion that includes that ordinal and that expansion must itself be expandable to all larger recursive ordinals.
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