Ouroborista

Then, one day, there was a man. A strange man in a strange robe on an always-identical fraction of time which normally did not house him. The intruder smiled, leaning up against a workbench, and it seemed like an expression he’d lost practice in. “You’re one lucky idiot, you know that” not the most polite greeting, but the best in a while. Bennet stayed silent, frozen in his corner like always and welcoming the fact that he had finally gone insane. “Oh have it your way. I can however assure you that forever does not get any more interesting.” he chuckled “Well– I suppose it does, but not without my help. Toodles.” Another fraction of infinity passed. Billions of identical seconds, perhaps trillions. Enough not to bother counting. On the billionth – perhaps trillionth – the odd man in his robe was back and introduced himself as Sisyphus. He appeared to have reacquainted himself with the practice of smiling. Either that or Bennett had simply grown more starved for the expression.  This time the prisoner of eternity answered. “So what did you mean? Last time?” A pause which seemed spiteful, then Sysiphus clapped his hands together and laughed as though this were the best joke he had ever heard. “Oh you will love this. Occasionally (some infinite number of times every arbitrary interval) I go through the records to see if anyone made any sort of interesting contract. If anyone managed to be clever... You, son, were not clever. You were lucky. Incredibly lucky, and perhaps, if there’s anything at all between those ears of yours, I can get you out of here.” A grimace unfurled itself in the centre of Bennetts face. “I appreciate the sentiment, but the contract literally says that I’m here forever” Slightly annoyed by the back-talk, the stranger tapped his cheekbone. “Not quite so. Those were going to be the rules, but you, smug douche that you are, asked the demon an incredibly stupid question: “How long’s that?” And they, being just as stupid and just as smug, gave you a harmless rephrasing, or at least something they thought to be a harmless rephrasing “Just count up to infinity, can’t possibly take that long”. The fell creature had an insufficient grasp on history, or perhaps it believed one of its more miraculous turns to be a mere fairytale. Who knows. I do not care and will not ask, but those beings who enjoy binding mortals to their prisms of chronology, they used to have a name for me: The eidolon of the lemniscate”. “Ah yes… I have no idea what that means” “You will. Just let me tell you a story first: Once there was a great king. Already he had been legendary amongst mortals for cheating death, and soon he was to become legendary too amongst those who are not. Hades did not take kindly to our hero’s shirking of expiry-dates and so he gave a task befitting of the undying. An endless punishment for a lesser man, had he not made the same error as your little critter. Hades forced the king to forever push a boulder up a hill. Each time it would roll down and he would start anew. An infinite number of times.” “I’m familiar with the tale.” “Not with how it ends, I assure you.” “Because it doesn’t end.” “False. It took me exactly 1.200 years to serve my infinity. With my help you can probably do even better than that” From his pocket Sisyphus produced a piece of chalk and began to draw on the walls of temporality. “You see, at first I went about my task mindlessly as anyone would, but time cannot help but force one to think eventually” He gave Bennett a somewhat pitying look. “For... for some people eventually just takes a while longer.” The eidolon of the lemniscate tried to give a smile midway between comforting and condescending and failed miserably. “It began with a thought about infinity and concluded just the same. The number of finite integers is boundless, is it not? Increment any number by one and the result will forever still be finite. No transfinite number is one larger than a finite one. But this presents an issue, does it not? Because in a very real sense integers count themselves. The first natural number is one. The second natural number is two. etc. etc. etc. If we have infinitely many finite numbers then, they too would have to count themselves and therefore the infinitely-many-eth would have to be infinitely large, thus not in fact finite. I admit I was troubled by this for a couple of years, but it was a productive frustration as I continued to think about mathematics. A thousand years had passed and I decided a very simple thing: An infinity of rock pushes could be subdivided into an infinite number of finite tasks. An infinite number of single rock pushes. I would give myself a hundred years to push it up once more. This was of course absolutely excessive, but it freed up more time to consider my plan and allowed me to save up strength. The next single rock push would be executed in half the time. The one after in half that still...” He drew a spiral pattern of the floor along with a formula: 100/1 + 100/2 + 100/4 + 100/8 + 100/16 … = 200 “And so it was done. I grew stronger and faster, and while I never stopped pushing the boulder up that hill – there never stopped being a next time – eventually, the infinity had passed.  Infinite tasks can be folded into finite space so long as you’re comfortable with geometric acceleration. Only later did I learn of Zeno’s paradox and I must admit that it made me chuckle to be so vindicated. It tells of Achilles and a tortoise with which he must catch up after it has already gained some distance. In the time it takes our hero to reach the spot occupied by the animal when he started, it has already moved to a further point. Once he reaches this new destination, time has again passed, and so the shelled adversary has progressed yet again. This one step is repeated an infinite number of times no matter how much faster Achilles is than the tortoise, because it will always move some finite distance in its finite time. Movement is impossible. Or so it might seem, but movement is not impossible. The infinite number of finite intervals follows a progression which makes it fit snugly into finite space. Achilles accomplishes his infinite number of tasks and catches up in a manner of seconds. Perhaps he does not even notice. All you have to do is pick some finite number, pick some finite interval in which you seek to get there – be generous – and then cover the next same–sized fraction of number space in half the time. Achilles always catches up.