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Expanding Mathematics

We can continue to expand the hierarchy of meaningful properties of the integers but not by considering more alternating quantifiers on the reals. The natural extension is to consider TMs that are second order well founded. This means they are well founded not for infinite sequences of integers but for infinite sequences of first order well founded TMs as described above.

The property of being a well founded TM allows us to define and do induction on recursive ordinals. Recursive ordinals support the definition of powerful general forms of induction. No formal system can enumerate all recursive ordinals. However the property of being second order well founded allows us to do induction on processes for generating recursive ordinals. This can be extended in obvious and very subtle ways. We can use the structures we generate to provide notations to index more powerful forms of extending the definition. In standard set theory all these notations are ordinals. The concept of ordinal masks the complex combinatorial structure of what are recursive processes for generating other recursive processes of a given type. In limited ways this approach is a return to the explicit typology of Principia Mathematica. The big advantage we have today is the aid of computer technology to manage and experiment with the complex type hierarchy that is generated by this approach. We can construct the computer programs being typed and experiment with their properties.

The inductive strength of Creative Objectivism could eventually exceed that of ZF. This philosophy encourages computer models and experimentation as aids to developing mathematical intuition. That may have the potential to extend mathematics in ways that are beyond the reach of current approaches. I suspect that eventually the language of ZF will prove inadequate to describe all the mechanisms for iterating the idea that notations defined at one level of mathematics can be used to index the construction of new notations.

ZF is formlated as a language describing external entities. As Gödel proved one can model ZF wihtin ZF but constructing the model is a complicated exercize. A more natural approach with this philosophy is to have a language that explicitly recognizes that it is talking about itself. The notation schemes we define in the language are used to extend the language. Everything is a recursive process but the typology of the inputs and outputs of these processes vary and the language of the formal system defines that variation.


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