A layman’s rant on infinity

(Written circa 2008)

I’ve always been fascinated with the concept of infinity since young. Long ago I’ve been pondering questions like the meaning of divisions by zero, arithmetic on infinities, recurring decimals (did you know 0.99999…. = 1 ?), etc. Most questions here answered by high school mathematics, yet many more questions came up in the course of events.

I learned, for example, that there were at least two types of infinities, one overwhelmed by the other. The countables and the uncountables. The Integers and Rationals on the one hand, and the Reals on the other.

Might be apt to say that my encounter with the Informatics Olympiad stuff brought me to another class of numbers which traditional mathematicians might not have put heavy emphasis upon – the Computables.

Did I say I was a layman? I only learned the names, the concepts, the ideas, but never got my hands dirty. Proofs I know only a select few, the more famous ones, like the Cantor diagonal argument. But I never had the knack for detailed mathematical work, and all those symbols seem weird and awkward. I really rather mess with words, or at least pseudo-words.

Yet the lack of knowledge into details never stopped me from thinking about infinities, or being fascinated. What are these things actually? In Mathematics we rarely invoke their full might in calculations. Inductions, Limits to infinity, Infinite sequences, etc…. all are techniques to summon the power of infinity without actually bringing their full force to the table. In limits and sequences we speak of “very, very close” (colloquially) and in inductions we say “and then it goes on forever and ever”.

We do set ourselves a boundary from the infinite (or infinitesimal), yet we peer over the border to achieve our goal. Yet … I ask, what lies beyond?

Here, to me, things start to get a bit philosophical. One thing I’ve always wondered is the “reality” of so called “Reals”. Despite their names, they really are rather elusive creatures. In fact, as a direct corollary to their uncountable nature, we cannot even name them all (unless you’re comfortable with potentially infinitely long names). It’s Pythagoras again, with their doubt of Irrationals, only this time we are witch hunting for the “undefinables”.

What lurks behinds these shadowy numbers? You may think that with our expressive language and grasp of Mathematics these little beasts are outnumbered by the numbers in the light, but in fact the numbers which are known to mortals are infinitely outnumbered. The power of uncountables is, as mentioned, overwhelming, and mortals only know how to count. We might have fancy methods of counting, fancy shorthands and other pretenses, but in the end all knowledge that we can put to words can be put into a countable set. The continuum hypothesis (and debate), though I never really understood or in fact learned it, seems to be yet another spooky thing.

Worse, as in the past few decades we’ve seen, that there are numbers that we know the precise definition yet are (seemingly?) unable to give a precise value. The halting problem fascinated me as much when I first heard of it, and things started to look creepy as when I learned about the incomputable busy beaver functions. The classical halting problem looks like a cute logic trick, without any hints infinities involved – the reason of the incomputability seems to be a fault of logic, that we were asking for an invalid task, as if we were asking an omnipotent being to create a stone so heavy that he could not lift it himself.

Yet the busy beaver functions suggests otherwise – the functions are one of the fastest growing functions known to mathematics, and they seem to link incomputability with largeness… as if there were some intricate connection between incomputability and infinity. I might sound like those people in more primitive (mathematically!) cultures that have a counting system like “one, two, many”, but the numbers that the busy beaver functions churn out seem to be so big that they might as well be infinite…

And then I keep asking myself… is there a Godly power in all this? With the different levels of infinities, would it suggest that different “Gods” might have different levels of powers? The “incomputably large” God may have power over mortals like us, but would it be subjected to the “countably infinite” God, who would then submit to the “uncountably infinite” God?

And why not? There are many many things you could do if you had power to overcome incomputability. (and many things you could not)