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There are two senses in which one mathematical system can be stronger than another: decidability and definability . Ordinarily the most powerful methods for deciding mathematical questions involve indirect methods operating on abstract structures. However one can artificially construct systems that are weak with respect to definability but strong with respect to decidability.
Zermelo Fraenkel (ZF ) set theory contains the axioms that are the most widely accepted foundation for mathematics. The axioms are comparatively simple and have enormous power. See Appendix on page . Virtually all of mathematics is formalizable within this system and virtually all proofs can be done based on these axioms. The power of these axioms come from the complex infinite sets that can be defined with them and the powerful iterative processes that can be defined to operate on these sets.
The power of ZF depends on the large infinite sets or cardinal numbers that can be defined it it. I doubt the existence of the hierarchy of cardinals numbers in ZF. From the Lowenheim Skolem theorem
we know that any formal system that has a model has a countable model .
It is possible to construct a formal system that speaks only of recursive processes that would be stronger than ZF in terms of deciding halting problems or the properties of nondeterministic TM s that we have described? The objects in such a system have a concrete existence. We can write computer programs that model them and we can use computer simulations to test our conjectures about their properties. It is conceivable that formal systems constructed in this way might be considered more certain than ZF and still be strong enough to prove the consistency of ZF.
The ordinal numbers are the framework of mathematics. If you define ordinals as infinite sets then you can always construct the next largest ordinal as the union of smaller ones. This is very powerful. It is too powerful. It allows you to wash over the rich combinatorial structure that evolves if you try to develop the same concept but stick to recursive structures. It is a detailed understanding of this recursive structure that will ultimately allow us to go beyond ZF with some confidence that we know what we are talking about. We are never going to accomplish that with reasoning about large cardinals.
One can give an outline of the functional hierarchy that needs to be developed. Start with a notation for recursive ordinals that allows one to recursively decide the relative size of two ordinals. No fixed notation can have this property for all recursive ordinals so the notation must be expandable. Use this notation as labels in a functional hierarchy. Level 0 in the hierarchy is the integers. Level N contains functionals well founded for objects at level (N-1). Objects at a limit level in the hierarchy are well founded for objects at all lower levels. You need to have explicit recursive labels whose ordering is recursively decidable for the input and output parameters so you know recursively what arguments are legitimate.
You need to generalize these ideas in strong ways while always working with recursive operations on recursive structures. How do you do this? There is a next major level in the hierarchy where the full structure I outlined above is only the successor operation . You can keep repeating this in some sense but how do you do it in strong ways and still always operate on recursive structure? There is of course no general answer to this question.
The ordinals and cardinals in ZF set theory define powerful structures like this implicitly. One must construct a candidate notation for recursive ordinals and prove within ZF it has the desired properties. One must then work out the detail of doing and generalizing induction on these structures. Such research is done not with the idea of going beyond ZF but to understand these structures. Mathematicians generally believe that it is too difficult to extend such structures to capture the power of ZF. As long as everyone working in this area believes this it will likely remain a self fulfilling prophecy.
I doubt that we can capture the power of ZF with conventional pencil and paper techniques. The recursive notations for recursive ordinals will become too complex. We need to implement the notations on a computer. We need to write programs to carry out iteration and we need to play with these programs trying different approaches to defining and generalizing iteration schemes and ordinal notations. A first approach at doing this is described in Appendix . Playing the game this way puts less emphasis on intellectual skills and more on intuition.
I believe this process will in the not too distant future lead to systems that are for more powerful than ZF in terms of decidability and that we will have more confidence in than we do in ZF. These systems will not be based on a questionable hierarchy of infinite cardinals. They will be based on computable iteration . Admittedly this iteration will be complex and that will cast some doubt on the consistency of the systems. There is a point where the level of recursive iteration
exceeds the capabilities of the human mind to comprehend. That is one of the implications of Gödel 's theorem if the mind is a computable process. I think we are far from that point. The axioms of ZF are short and simple. With the right techniques I think we can comprehend the combinatorial iteration techniques implicitly defined by those axioms. In the process I think we will come to understand how to define much more powerful iteration techniques. I believe this understanding will enrich mathematics and have enormously important implications for science and technology.
Alternating between implicit and explicit definability in mathematics is another of the cycles that drive creativity. I am advocating a return to the approach of Principia Mathematica [35,36,37,38] where the hierarchy of types was explicitly enumerated. The two improvements I am suggesting are the use of recursive function theory (a theory that did not exist at the time of Principia Mathematica) to simplify the construction of the typed hierarchy and the use of computers (which did not exist then either) to get a hold on the complexity of the task.
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