**Infinity as a concept in mathematics is simple to
describe. It is conceived as a dichotomy. There is a potential infinity. This
is the infinity that most mathematicians accept and use. It posits that
infinity is only a potential thing we can know through analytic methods. Take
the endless augmentation of numbers. Numbers go on forever because we have a
set that increases when coupled with a rule to expand them, thus we can never
obtain the last member of this set. Yet, we never experience this ever going
augmentation of numbers, it is only potentially infinite for us. It is never
an actual infinitude. All that set theory mathematics can guarantee us is if
we follow the rules of augmentation for a set of primitives (i.e. objects) we
will never obtain a last member of a given set. So this leads to the second
notion of infinity: actual infinity. In this idea there is a real existing
infinity that we need not posit as a possibility. It really exists. Proponents
of this concept, known as Intuitionists believe that there is a real existing
infinity, though we can't perceive it.But, there is a third type of infinity
I will explore that dwarfs both these concepts and is often ignored. It is the
idea of a unique infinity, which may just not exist at all. I mean it may not
exist in our physical world, but must be at least abstractly possible.**

** **

**Unique infinity is a collection of objects none of which
are the same as their antecedents. So taking numbers, every number in this
infinity could not be a composite of the other numbers. Moreover, prime
numbers don't have this uniqueness property. Prime numbers are in some sense
combinations of other numbers. For instance, both 7 and 17 are prime, but 17
is really 10 + 7, which makes use of a previously named number. All that
primes require is that they are not factors of any other number but themselves
and 1. This requirement isn't strong enough to make them into unique infinite
sets. For a number to be truly unique, it has to be completely different from
its antecedents. This would amount to ever-changing names or shapes for all
numbers. We couldn't augment them if this were the case. Imagine having the
number one billion formed by constantly changing symbols.To name the number
we would have to have one billion different symbols strung together. The
system of augmentation that uses a binary operation on symbols that recur with
their placement value shifting to the right wouldn't work. But still, there
should be a set of unique objects none of which can be derived from the
others. This kind of infinity is arithmetically impossible with our system of
numeration now, but theoretically possible. Or is it? Reductionism teaches us
that all things are reducible to a limited number of constituent parts. The
limited number of constituent parts can be combined via some rule into very
complex and endlessly varied things. So, take language as an example. We can
combine a limited number of symbols to form an endless number of words and
sentences. Even though some of those words and sentences will be nonsense,
still they will be infinite and generated from the finite. A good example can
be found genetics. With four basic genes, seemingly endless forms of unique
life forms can be derived. Now consider if we had continual uniqueness in
every single possible element of either of these systems what would happen.
The answer is these collections of objects would never achieve any system and
could never become larger and more complex combinations of simpler forms. This
is the same as saying no language or any life would exist, if everything that
composed it was unique. It seems in order for infinity to be actualized beyond
the conceptual arena, we must have limitation in what composes it. Paradoxical
to say the least, isn't it? We can ask ourselves the following strange
question:**

**Why must there be limitation of objects to create
limitlessness in augmentation?**

**The answer to this question is one of those yes
and no constructs.**

**Yes, there is a conceivable unique infinity, in which
everything it embodies is always different from everything else. No, this type
of infinity doesn't exist in the physical world. The more interesting
speculation is what kind of conceivable infinity would this be? Just think of
it, a collection of objects such that none is in any shape, form and fashion is
not similar to that which precedes it! It defies our nice, familiar way of
conceptualizing.We need the finite to build the infinite.Yet, the infinite
with no reference to the finite is mind-boggling.**

**I would like to think in the reverse of the above line of
reasoning. Is it possible that we can start with uniquely different things and
they decompose to similar things and thus our kind of infinity emerges?**

**A Statement That Can't Be Proved:**

**Given we have a unique infinite set called {X}, where
represents any set of unique objects, then {X} will become a set {Φ},
where Φ represents objects combinable by an operation rule that acts upon
a limited number of {X} objects. **

**The problem with the above assumption is we would never
know how set {} becomes {Φ}.There is no way to derive one from the
other in the assumption and thus it becomes invalid. If we could perform this
leap, and then much would make more sense in our physical world. The Big Bang
theory would achieve a new meaning. We could see the initial infinite density
point of the universe as becoming {Φ}. That is, we started with a
pre-universe not of a unity of 4 basic elements: strong force, weak force,
gravity and magnetism, but one in which there was an infinity of unique things
that congealed into limited, but infinitely combinable things. **

* *

*Robleh Wais*

*11/26/10*