Saturday, February 11, 2012

Are some infinities are bigger than other infinities?

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!

Paula Deen and No Edge

First off, quick fact. This picture of Paula Deen riding John Green is not the first time. When John Green's book Paper Towns peaked in NYT bestseller list at #3, Paula Deen's cookbook was one of the two books that were top of John Green's book. So there.

Now on to the main topic: NO EDGE. To talk about the edge of the universe, you have to talk about the shape of the universe. Now best way to illustrate this is through an analogy with a rubber sheet. At one time, our rubber sheet was a clumped ball, consisted of everything that was there. Mass, time, space... everthing was there in that tiny dot. But then that dot slowly grew until a great force within spread the universe like a big kablooey. Thus came the two terms of this event, the big bang and the horrendous space kablooey. Now that great force had a effect like a sharp tug, as you know flattens the things you tugged. So the light shone in empty space will not come back to you but come out in a straight line, unless there's stuff, where gravity curves the space-time. Since the curve is unnatural, curves want to meet together so to become a overall flatter shape.

Now I may lost you in explaining how gravity works, but here comes the deal. The battle between the 'tug' or expansion and gravity is what determines the shape, gravity wants to curve so much as to disappear, you have a big crunch where we start to collapse back on itself and space-time goes haywire. Maybe 'tug' or let's call them what they really is, dark energy, tugs too much and starts to rip as the fabric can't take it anymore and we experience what is like a 10th Doctor's regeneration only with everything and we don't end up becoming a swanky British man with a bow-tie and sometime a fez. That's called a Big Rip. Maybe gravity will win slightly win curve to a sphere-like shape where time and space might repeat itself forever,like a big bounce, or maybe dark energy will win slightly and will spread forever resulting in heat death or a big freeze. Big bounce is actually a term when there's fluctuation between gravity and dark energy and we go back from big bang to big crunch repeatedly in a different fashion. (so this is like a Doctor's regeneration) Now in a big crunch or a big bounce or maybe a big freeze, a universe curve back to itself, therefore effectively having a no edge like a surface of a sphere, where you could go in any direction and come back to point you just started. But if it's otherwise and universe is really a flat surface or a hyperbolic saddle shaped there might be an edge if the universe we see is finite and it is so... (live. I mean universe which we live, not see. There's a difference in that.) Anyway, in conclusion, we don't know if there's an edge of the universe or not. Sorry.

[Actually universe might even ALL EDGE. As in we're just a holographic projection of a 2-d world.]