The Finite Infinity

My first real brush with infinity came during an interesting discussion within one of my Calculus classes. The teacher asked us, "what is infinity?" Well, on the surface, the question seems so simple as to be a loaded question. In fact, it would soon prove to be after I answered my professor.

"Infinity is a limitless bound -- an arrangement of things that continues long after we have ceased to continue counting."

"Fair enough," said the professor. "However, are there different sizes of infinity?"

Now the fact that the original question was indeed loaded became apparent. I knew where he would go with this line of thinking.

"Take for instance the collection of all natural numbers. How many are there?"

"An infinite collection," I answered.

"Alright then. Let's call the collection of all natural numbers an infinite set. If this is the case, how many even numbers are there?"

"An infinite number of even numbers," I answered.

"If the set of all natural numbers is infinite, and even numbers are a subset of that infinite set yet this subset is ALSO infinite, does this suggest that one infinity could be smaller or larger than another?"

I couldn't answer him immediately. I realized that the subset of all even numbers within the set of natural numbers is indeed infinite, but that would mean the set of natural numbers was twice as infinite since it also included odd numbers as well. My mind had trouble dealing with the concept that there could be an infinity twice as large as another one yet both were infinite.

If we stopped here and "pretended" that the set of all natural numbers included only the numbers one through 10, then we could show that this set had 10 members. From this limited set, we could easily show that there was a subset of 5 even numbers within it. It would be very clear that this subset contained half the members of the total set of all natural numbers.

Yet we know that the true set of natural numbers continues without end so long as we continue to count higher.

Although this is true, we could dive deeper into the problem and show that mathematics, as a whole, is an abstraction of reality. What possible meaning could a very large number "X" contain if "X" was greater than any countable thing in the universe? There are only so many elementary particles in our universe. The Planck constant argues that space and time are only divisible up to a certain point until there is no smaller measurement of space and time. Knowing this, one could argue that at some point, one would count high enough to reach a number that had no correlation or meaning to the universe which brought it into existence since there was no real countable thing to represent that number. At this point, it would become a true abstraction of "nothing" -- just another intellectual musing without any real foundation.

If X is the total collection of all things within the universe, X+1 becomes a number outside any tangible way to express that number within a universe that doesn't have enough basic elements to give X+1 any true meaning.

At this point, infinity begins to break down. Infinity's little brother would be the "infinitesimal." Yet the Planck constant forces us to acknowledge that, in the real world, things cannot really continue without end to become ever smaller. In an abstract sense, sure -- but in reality? No.

So the concept of infinity strays into the abstract once all real representations of countable things is exhausted. It becomes just a number for a mathematician to play with. In an abstract world, you could very easily have different sizes of infinity, but without any basis in the concrete countable world, those different sizes of infinity just become an academic curiousity.

Amazingly, you could have an infinity that was infinitely larger than another infinity. If the set of all natural numbers is infinite (in an abstract sense), then one could say, "take every infinite number within this set and create a subset from them." You'll never get to the first member of that subset, yet in an abstract sense there is still an infinite number of them -- you'll just never come across one.

Infinity divided by infinity is undefined without knowing more about the functions that created those abstractions. Infinity (members in the set of natural numbers) divided by infinity (members in the set of all even numbers) should equal two, right? Maybe?