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2 Background

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The most widely used formalization of mathematics, Zermelo Frankel set theory plus the axiom of choice (ZFC)[3], gives the same existential status to every object from the empty set to large cardinals. Finite objects and structures can exist physically. As far as we know this is not true of any infinite objects. Our universe could be potentially infinite but it does not seem to harbor actual infinities.
This disconnect between physical reality and mathematics has long been a point of contention. One major reason it has not been resolved is the power of the existing mathematical framework to solve problems that are relevant to a finite but potentially infinite universe and that cannot be solved by weaker systems. A field of mathematics has been created, called reverse mathematics, to determine the weakest formal system that can solve specific problems. There are problems in objective mathematics that have been shown to be solvable only by using large cardinal axioms that extend ZF. This however has not resulted in widespread acceptance of such axioms. For one thing there are weaker axioms (in terms of definability) that could solve these problems. There do not exist formal systems limited to objective mathematics that include such axioms1 in part because of the combinatorial complexity they require. Large cardinal axioms are a simpler and more elegant way to accomplish the same result, but one can prove that alternatives exist. Mathematics can be expanded at many levels in the ordinal hierarchy. Determining the minimal ordinal that decides some question is very different from determining the minimum formal system that does so.

2.1  The ordinal hierarchy

Ordinal numbers generalize induction on the integers. As such they form the backbone of mathematics. Every integer is an ordinal. The set of all integers, ω, is the smallest infinite ordinal. There are three types of ordinals: 0 (or the empty set), successors and limits. ω is the first limit ordinal. The successor of an ordinal is defined to be the union of the ordinal with itself. Thus for any two ordinals a and b a < b ≡ a ∈ b. This is very convenient, but it masks the rich combinatorial structure required to define finite ordinal notations and the rules for manipulating them.
From an objective standpoint it is more useful to think of ordinals as properties of recursive processes. The recursive ordinals are those whose structure can be enumerated by a recursive process. For any recursive ordinal, R, on can define a unique sequence of finite symbols (a notation) to represent each ordinal ≤ R. For these notations one can define a recursive process that evaluates the relative size of any two notations and a recursive process that enumerates the notations for all ordinals smaller than that represented by any notation.
Starting with the recursive ordinals there are many places where the hierarchy can be expanded. It appears that the higher up the ordinal hierarchy one works, the stronger the results that can be obtained for a given level of effort. However, I suspect, and history suggests, that the strongest results will ultimately be obtained by working out the details at the level of recursive and countable ordinals. These are the objective levels.

2.2  The true power set

Going beyond the countable ordinals with the power set axiom moves beyond objective mathematics. No formal system can capture what mathematicians want to mean by the true power set of the integers or any other uncountable set. This follows from Cantor's proof that the reals are not countable and the Löwenheim-Skolem theorem that established that every formal system that has a model must have a countable model. The collection of all the subsets of the integers provably definable in ZF is countable. Of course it is not countable within ZF. The union of all sets provably definable by any large cardinal axiom defined now or that ever can be defined in any possible finite formal system is countable. One way some mathematicians claim to get around this is to say the true ZF includes an axiom for every true real number asserting its existence. This is a bit like the legislator who wanted to pass a law that π is 3 1/7. You can make the law but you cannot enforce it.
My objections to ZF are not to the use of large cardinal axioms, but to some of the philosophical positions associated with them and the practical implications of those positions[1]. Instead of seeing formal systems for what they are, recursive processes for enumerating theorems, they are seen by some as as transcending the finite limits of physical existence. In the Platonic philosophy of mathematics, the human mind transcends the limitations of physical existence with direct insight into the nature of the infinite. The infinite is not a potential that can never be realized. It is a Platonic objective reality that the human mind, when properly trained, can have direct insight into.
This raises the status of the human mind and, most importantly, forces non mental tools that mathematicians might use into a secondary role. This was demonstrated when a computer was used to solve the long standing four color problem because of the large number of special cases that had to be considered. Instead of seeing this as a mathematical triumph that pointed the way to leveraging computer technology to aid mathematics, there were attempts to delegitimize this approach because it went beyond what was practical to do by human mental capacity alone.
Computer technology can help to deal with the combinatorial explosion that occurs in directly developing axioms for large recursive ordinals[2]. Spelling out the structure of these ordinals is likely to provide critical insight that allows much larger expansion of the ordinal hierarchy than is possible with the unaided human mind even with large cardinal axioms. If computers come to play a central role in expanding the foundations of mathematics, it will significantly alter practice and training in some parts of mathematics.
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Paul Budnik. What is and what will be: Integrating spirituality and science. Mountain Math Software, Los Gatos, CA, 2006.
Paul Budnik. A Computational Approach to the Ordinal Numbers. Mountain Math Software, Los Gatos, CA, 2010.
Paul J. Cohen. Set Theory and the Continuum Hypothesis. W. A. Benjamin Inc., New York, Amsterdam, 1966.


1Axioms that assert the existence of large recursive ordinals can provide objective extensions to objective formalizations of mathematics. Large cardinal axioms imply the existence of large recursive ordinals that can solve many of the problems currently only solvable with large cardinals. However, deriving explicit formulations of the recursive ordinals provably definable in ZF alone is a task that has yet to be completed. With large cardinal axioms one implicitly defines larger recursive ordinals than those provably definable in ZF.

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