The Hilbert Hotel has remained, to this day, one of my favorite paradoxes of all time, out of the ones I have learned and read about in the past as a curious, young kid. It has also remained, to me, as one of the more intriguing scenarios thought up by mathematicians, theorists, and scientists, that, by its very nature, seems to defy logic and common sense. Strogatz’s article on various aspects of the Hilbert Hotel overall interested me very much, but what I found extremely interesting was his implications of there being multiple types of infinity, and that some infinities are larger than other infinities (something I already knew before hand from watching Numberphile and Vsauce on Youtube). I find it mind boggling that the number of real numbers and their positive fractions are actually larger infinities than can be accommodated by the Hilbert Hotel despite it always claiming that there’s “always room”. (Funnily enough, this proves that even in the realms of paradoxes and pure mathematical theory, customers and investors can still get scammed and ripped off. Corporate cronyism is ubiquitous). But this makes sense, as the theoretical number of real decimal numbers between any two given adjacent values is…infinite– simply because of the fact that a repeating decimal can go on forever, and there is always potential to add another number to the infinite decimal sequence. Even if you think you’re done at 0.99 (say, if you were trying to find all the real decimal numbers between 0 and 1– as futile as the task would be), there’s always 0.999, 0.9999, 0.9999999999999999999999, and so on and so forth ad infinitum. What I didn’t quite understand was when Strogatz, in his article, postulated a scenario in which every real number got its own room, and when he said you could change any number in each decimal as shown in the diagram and get a different decimal. He lost me when he wrote: “This new decimal .325… is the killer”. I know it has something to do with infinity (duh!), but I wasn’t sure if he was talking about combinatorics or an infinite set or whatever. After reading Strogatz’s article about the Hilbert Hotel, I myself started asking questions about infinity and other paradoxes that could involve infinity– sort of like how the 6-year-old (Ben) was asking whether infinity was odd or even at the beginning of the article (which I found quite precocious for a mere 6-year-old. We need more people like him in this world, just saying). I eventually remembered some paradoxes I myself thought of in high school and had written down a long time ago, which after reading this article, made me think of their relation to infinity. I don’t know if both of them relate to infinity directly, but I’ll state them below:
- A student who aspires to become the best swordsman in all of Asia leaves his master in his monastery, who tells him that he is ready to prove himself to the world (all the while knowing of his student’s brashness and overconfidence). The student goes and fights the best swordsmen across the different nations and lands of Asia– China, Japan, Korea, Manchuria, Thailand, and more, and wins every single duel he has with them, claiming victory over his opponents and subsequently receiving praise from them for his skills. When he comes back the student tells his master that he is the best swordsman in all of Asia, and that no one can or will be able to defeat him. The master becomes enraged and tells his student that he has failed on his quest, while the student protests and stares at his master in shame and shock. He asks why he has failed if he had defeated the best of the best, but the master replies that he did not beat himself– his own arrogance and that in order to beat himself so that he can become the best swordsman, he must go back to all the masters he fought all around Asia and lose to them. For those masters knew the boy was brash and went easy on him, inflating his ego more and more! The student, ashamed and feeling like a failure, realizes his mistake and goes back to fight his opponents. The paradox is that in order to win he has to lose, and vice versa. You could think of every +1 (win) is now being replaced by a -1 (loss), meaning he will always end up numerically, with a net average win/loss ratio of 0, if that makes any sense. Alternatively, since he “wins” every time he loses to all the masters (-1 repeated), he “gains” a +1, meaning he always ends up with a value 1 greater than when he initially started (I think).
- A cheetah in the African Sahara is prowling in the tall grass when he sees a seemingly oblivious thirsty gazelle drinking from a lake in the middle of the plains. The gazelle somehow heard the cheetah prowling because of the rustling of the grass, and knows perfectly well the cheetah is there ready to pounce on him and maul him to death. However, the cheetah upon inching closer to the gazelle, knows that the gazelle knows that he is there, ruining his tactics of catching the gazelle off guard so as to surprise him and eat him right then and there. Since both creatures know of the other’s existence, this means that the hungry cheetah cannot pounce without the gazelle running away milliseconds after (recognizing his existence) and that the gazelle cannot run away unless the cheetah foolishly tries to attain the gazelle for his rumbling stomach. The gazelle is obviously not going to stay there drinking, knowing a predator is near, and the cheetah is not going to stay there in the grass, knowing his has a massive opportunity to fill his gut with gazelle meat. And yet both animals have to stay there (they must do nothing AND something at the same time). Theoretically, the only logical solution to this problem would be that both animals would have to stay in place for an infinite amount of time, but the problem with that would be that both animals would die long before then, and they would be the meal of the worms (or the vultures).