# 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
axioms^{1} 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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## References

- [1]
- Paul Budnik.
*What is and what will be:
Integrating spirituality and science*. Mountain Math Software, Los Gatos,
CA, 2006.
- [2]
- Paul Budnik.
*A Computational Approach
to the Ordinal Numbers*. Mountain Math Software, Los Gatos,
CA, 2010.
- [3]
- Paul J. Cohen.
*Set Theory and the Continuum
Hypothesis*. W. A. Benjamin Inc., New York, Amsterdam,
1966.

### Footnotes:

^{1}Axioms 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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