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