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An analogue to recursive notations for larger countable ordinals


This is part of a project to use computers to develop an explicit formulation of the recursive ordinals implicitly defined within Zermelo-Frankel set theory (ZF). There are two motivating ideas. First is the original approach to mathematical infinity that treats it as a potential that can never be realized. This suggest to me that objective mathematics is limited to statements which are logically determined by a recursively enumerable sequence of events and thus are relevant to an always finite but potentially infinite universe. This includes virtually all mathematics of practical significance including some instances of quantification over the reals such as that needed to define the recursive ordinals.

In this view reals are human created but can be objective. Cantor did not prove there are more reals than integers. Instead his famous proof is the first of the great incompleteness theorems. Cantor proved that one can always expand a formal system and define new reals by diagonalizing all the reals provably definable within the system. We know by the Lowenheim-Skolem theorem that these reals must be countable outside but not inside the formal system that defines them.

It is consistent to assume uncountable sets exist, because the set of all reals provably definable in set theory are a well defined collection. In an always finite (but possibly potentially infinite) universe, the uncountable sets have a 'through the looking glass' existence. They cannot be modeled by a definite something, but only by a collection that changes as mathematics expands.

The second motivating idea is the belief that a computational approach to the recursive ordinals will eventually go far beyond what is possible without using the computer as an essential research tool. Once we have an explicit formulation of the recursive ordinals provably definable in ZF, I think we will have the machinery to go far beyond ZF and far beyond ZF + large cardinal axioms in extending the combinatorial implications of ZF.

To learn more see the paper, What is Mathematics About?, the book What is and what will be and the video Mathematical Infinity and Human Destiny.

Admissible level ordinals

The ordinal calculator includes notations for what I call admissible level ordinals. These are countable ordinals ≥ the Church Kleene ordinal or the second admissible ordinal. (The first admissible ordinal is the ordinal of the integers, ω.) Higher level structures are often a good way to expand a system of lower level structures, in this case notations for recursive ordinals. This approach supports a form of ordinal collapsing that can be done at any countable admissible ordinal for which a notation exists in the system.

Generalizing recursive ordinal notations

Recursive notations for recursive ordinals can fully enumerate the structure of the ordinal they denote by, for example, enumerating notations for all smaller ordinals and providing a recursive method to determine the relative size of any two of these notations. Such a notation is impossible for the ordinal of the recursive ordinals and all larger ordinals.

However, by developing an explicitly incomplete notation system, one can construct an analogue of recursive ordinal notations for countable admissible level ordinals. In the notation for recursive limit ordinals in the current ordinal calculator, a recursive function on the integers (called 'limitElement') enumerates a sequence of notations for smaller ordinals which have this ordinal as their limit. If this function on integers is replaced with a function on ordinal notations (called 'limitOrd'), then one can duplicate the rest of the properties of recursive ordinal notations for countable ordinals ≥ the Church Kleene ordinal.

'limitOrd', for each admissible ordinal at level α + 1, accepts inputs of notations for all ordinals at all levels ≤ α. One cannot enumerate notations for these ordinal parameters of 'limitOrd' as one can the integer parameters of 'limitElement'. Thus this expanded notation system is incomplete. The program that implements these ideas is expandable using C++ subclasses and virtual functions.

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