**Classical Infinity**

**Infinity as a concept in mathematics is simple to describe. It is conceived as a dichotomy. There is 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 a 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
{§}, where § represents any set of unique objects, then {§} will become a set {Φ},
where Φ represents objects combinable by an operation rule that acts upon
a limited number of {§} 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. **

*Ken Wais*

*11/26/10*