The quote of some infinities are bigger than other infinities have been "the quote" of The Fault in Our Stars. But there's no conflict whether it's true or not. Someone already prove that. And I'm going to show you right now.
What is a number? How do we use numbers? Well, there's three of apples and 10 trillion (and counting) of US's debts in the units of dollars, so I guess we are talking about a set of stuff. So numbers will help you define a set of stuff. What about set of numbers. How many natural numbers are there? Well there's one, two, three and we could add one forever and ever... And so we give up and call that set infinite. How about integers? Well that's like natural number plus the negative numbers (and zero, don't forget the zero) so that's infinite. In fact, since every natural number has its negative number, it'll be doubly infinite. So the title question above, proved. Well, not exactly. Why do you say something is bigger? How do we know there's more stuff in one set then in an another set? We line them up and see which occupy more space. One that occupy more space are bigger set compared to the other set. So let's line those numbers up. 1,2,3... and below that 0,1, -1... Well, looks like they're going to infinity, since we can't get there, let's say they're exactly same amount... and... yeah. How about the rational numbers? Well, rational number is just ratio of two integers so it's like line of integers multiplied by a line of integers so it's like a square, and you can't fill a square with a line. What about space filling 'curves'? Oh, yeah. I forget that. So set of rational numbers are also equal amount as the natural numbers. And now the big question. How about real numbers? Oh god man, there's like infinite one between every rational number, and the actual number go off to infinity, man. I don't know what to. Well can you just list off anyway? With what? I don't know. Keysmash! Okay.
1 1.4973279805491585735125795871409709537832091842185811389501580328901582309851...
2 62345323621424.092384320958154908514649741978549083098918719173907891027905919...
3 340987213.20498420481574059832085098504981095803129891275907192873509874233928...
4 30948270948.350927518979071942873209573901783257091280957180293759872135901781...
5 32509892135791207580397592357380291578205975780924387.320958095809487703497614...
6 .20947832957928570982789017589057812097920139579487130690703970297960763209769...
7 32934289.571578578941047839238042987643739587439451873867479342524878483485243...
8 203579938.32875698732165093570195872978506835810256892306578234689156085832739...
9 4896893423205643305.2589857914872389031295746732878948697623598138491574387932...
Wait, I see something. Let's do this, make a number which the first digit is not the same as the first digit corresponded by the first natural number. Then make its second digit not be same as the second digit of second corresponding number, and so on and so forth...
Ok. 492725381.90385309275187897420195890175934875147710928759014781247809784093759...
So what's that have to d... Where does this number fit in? I don't know man, the number goes to infinite so I guess it'll come up somewhere. Where exactly? I don't know, a millionth? A billionth? What about that millionth digit or the billion digit? Does that have to be different from millionth number in the second column or the billionth? Well, that's true and the number goes off the infinity and the first list goes off to infinity so... oh. If you have set of boxes and set of apple and you put an apple per box and have apple left over, is the set of boxes bigger of set of apples? Of course,the apples... does that mean set of real numbers is bigger than set of natural numbers? Yes. And there you have it. The concrete proof that some infinities are bigger then other infinities. In fact by using this formula we can class of infinity all with different set value. We call these transfinite numbers and we could treat them as numbers beyond numbers and there is a number of set of all infinities. It's called omega and you could argue that is the biggest number ever imaginable, unless there's other omegas then watch as progression of infinity break the bounds of reality and you agonize in pain for just thinking it. So omega is the biggest number. So use that to your biggest advantage. 'infinite times!' 'infinite + 1 times!' 'No, that's still infinity.' 'How about omega time?' 'Darn.' Actually Georg Canter, who found all of this said that omega is like a god, and so using it in that way is taking god's name in vain, wouldn't it? Anyway, that's why I don't believe in a god. And good night!
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