**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 in 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*