Krystalzkris:
i feel lost here... Joshua can u help me out with this?@Killsmith

killsmith:
leave all these big grammer...the OP's summary is, concept of infinity...how the natural numbers and the even numbers are equivalent...and there are infinite real numbers between any 2 real numbers.....fear not

Teempakguy:


1. Hilbert's paradox of the Grand Hotel.

so, this asks, is it possible to fill a "Grand Hotel" which has an infinite number of rooms, with an infinite number of guests? well, i for one do not know. let's say you can. what then happens when a new guest arrives? there an infinite number of rooms nah, so obviously, the guest will find a room. but if he finds a room, then the hotel was never filled in the first place. but it is said there are an infinite number of guests. so, why can't they fill it?

2. The omnipotent being paradox.
i apologize in advance to theists. this might hurt a bit.

this paradox simply asks, is it possible for an infinitely powerful being, such as an omnipotent God, to create something so infinitely powerful that he/she cannot carry it?
if he can do so, then he/she just made something they cannot move, therefore, they are not infinitely powerful. if they cannot do so, well, they are not omnipotent either way.

socially4fun:
The concept of infinity is very tricky. What if I tell you some infinity is greater than some. Example, all prime numbers are a subset of all positive integer.

I can conclude saying, sum of all prime numbers are infinite, but are less than the sum of all positive integers.

Another instance, what if I add infinity plus infinity... will I get infinity or 2Infinity.

Let infinity = X
X + X = 2X
2X > X
2X = infinity
see the twist?
Infinity can be greater than infinity.


Thanks for the insight.

hahn:
Cool.

You guys just keep pointing out how dumb I am

Karmanaut:
The banach tarski paradox is a little more specific than that. A sphere is a set of point, and two sphere is also a set of point, those two sets have the same cardinality so there is a bijection between those two.
So you can completely cut a sphere apart (each point making a piece) and recombine the (many many) pieces in order to form 2 sphere, but that's not really interesting and the ancient Greeks already knew that there was as many points in a circle of radius 1 as there is in a circle of radius 2.
So the interesting part of Banach-Tarski is that the pieces are in a finite number (and 5 is that numer) and then can be recombined using only isometry to form 2 spheres. Finite and Isometry are important words here.
And if anyone wonder what is the trick in the Banach tarski paradox it's this : you'd think that it's impossible because it's against conservation of volume but the trick is that the pieces you cut the sphere in to are so complicated and so weird it's not possible to define volume for them. If they don't have a volume then the conservation of volume doesn't mean anything for those pieces and can be "broken".
So : a sphere has a volume, you cut this sphere in 5 specific pieces so strange that they don't have volume (separately), you move the parts so you obtain 2 spheres, those 2 sphere are not so strange and have a well defined volume.

llaykorn:
This is a very nice piece, Teempakguy; it takes me back to my childhood days. Daddy used to barb his own hair while we were kids and he would place one mirror behind him and the other in front of him. In that way, he could see the back of his head from the mirror in front of him. One day, I went into his bathroom and stood between the two mirrors and I was quite fascinated to realize that as the mirror in front of me reflected what was on the mirror behind me, the mirror behind me was also reflecting what was reflected on the mirror in front of me. It didn't stop there. The mirror in front of me again was reflecting what the mirror behind me was reflecting of it. Then, it didn't even stop there. I tried over for some days to count how many of the reflections the mirrors showed, but they always got too tiny for my little eyes to see. Hehe

The number of reflections on Daddy's mirrors was definitely infinite....

Teempakguy:
Hey guys!


i can see that's there's been some problem with the omnipotent paradox.
well, the truth is, i didn't make it up. it's a pretty common paradox in philosophy and sometimes mathematics and is usually used to illustrate the impossibility of infinity occurring in the real world. i am going to give an in depth explanation here.

Teempakguy:
Hey guys!


i can see that's there's been some problem with the omnipotent paradox.
well, the truth is, i didn't make it up. it's a pretty common paradox in philosophy and sometimes mathematics and is usually used to illustrate the impossibility of infinity occurring in the real world. i am going to give an in depth explanation here.

cc abiambo, Nmeri17, ELYMAXiimus.

---

let us find a very powerful being. so powerful, that his power is infinite. which means he can do anything and everything. we will name him Bronut.

now, Bronut is asked to create a very dense and heavy box. so heavy that he cannot carry it.

the question now is, can he?

does Bronut have the ability to make a box so heavy that he cannot carry it?

1.if bronut can make this box, he will not be able to carry it. and if he cannot carry it, that means his power is limited. not infinite. he cannot do anything after all.

2. if Bronut CANNOT make this box, then is also means that he cannot do everything. His power is not infinite because he cannot do everything as he earlier claimed. including making a box he cannot carry.


another version is, is it possible for an infinitely intelligent entity to create a problem he cannot solve? if he can create such a problem, then we know he is not smart enough to solve his own problems. if he cannot create a problem so difficult that he cannot solve it, then his smartness is not unlimited then.



i hope you got that . . . for more official explanation, LOL, go to wikipedia. HERE.

Teempakguy:
Hey guys!


i can see that's there's been some problem with the omnipotent paradox.
well, the truth is, i didn't make it up. it's a pretty common paradox in philosophy and sometimes mathematics and is usually used to illustrate the impossibility of infinity occurring in the real world. i am going to give an in depth explanation here.

cc abiambo, Nmeri17, ELYMAXiimus.

---

let us find a very powerful being. so powerful, that his power is infinite. which means he can do anything and everything. we will name him Bronut.

now, Bronut is asked to create a very dense and heavy box. so heavy that he cannot carry it.

the question now is, can he?

does Bronut have the ability to make a box so heavy that he cannot carry it?

1.if bronut can make this box, he will not be able to carry it. and if he cannot carry it, that means his power is limited. not infinite. he cannot do anything after all.

2. if Bronut CANNOT make this box, then is also means that he cannot do everything. His power is not infinite because he cannot do everything as he earlier claimed. including making a box he cannot carry.


another version is, is it possible for an infinitely intelligent entity to create a problem he cannot solve? if he can create such a problem, then we know he is not smart enough to solve his own problems. if he cannot create a problem so difficult that he cannot solve it, then his smartness is not unlimited then.



i hope you got that . . . for more official explanation, LOL, go to wikipedia. HERE.

rhizoidz:
Like a scientist I love being opposite, questioning the root of things and ideas before trying to get answers,

I'm going to relate the omnipotence paradox to a mathematical theory of the idea of the rotation of the moon around the earth.

Instances
1. X is omnipotent, which is defined as the ability to do all.
2. Can X create an entity(Y) which is too great for him to destroy?

Possibilities
1. Just like space and time would be opposite sides of the same entity, creation and destruction, life and death, infinity and zero are opposite sides of the same entity, the fact that for X to be omnipotent, he must be able to do Y and undo it at the same time,each of which makes the other impossible, which is supposed to be normal , one observation of mine is that whenever two equal sides if the same entity are imposed , they create a continuum, which both are present and at the same time none is present.

2. The factors that would confirm X's nature of being omnipotent is the ability of X 1. Be able to create Y which he should not be able to destroy then 2. Destroy Y, if he can do both, then he is omnipotent, isn't that so? He has to obey two forces of the same magnitude but different directions, now relate this to the rotation of the moon about the earth... The moon is subjected to obey the law of gravity and that of the first law, so did that mean a massive explosion? No an infinite loop of cycles, then if X's ability to create Y and then destroy it made him omnipotent, it might have also made him eternal, stuck in a loop of cycles...
Darn it!! I've just created another question!

Looking up to you peeps, can you help me sort this hypothesis, and put it in the right order.

ValentineMary:
Nice work Tempakguy. Really enjoyed it. Reminds me of an IQ test I took once and they asked how many chocolate(s) would it take to fill an infinitely empty stomach. And I was told d ans is 1 pls can u explain how they arrived at this value to me. Thanks

Geofavor:
. Teempakguy

hey there, guys . . .

i recently received a mention from the dairies section, urging me to visit the science section more often, and at this time i was in the art world.

so, here i am and . . . i have a rather fascinating topic I've been thinking about and doing thought experiments about. and er . . . yeah, it's quite an intruiging one. ALTHOUGH, it's is a very mathematical topic. so, put on your thinking cap on, guys

So what is infinity?

well, the concept of infinity first occurred to me when i was a child. about er . . . five or six . . when, in the toilet, . . . i saw a bottle of harpic toilet cleaner on the wash basin and noticed that the bottle had a picture of itself on it. as in, on the bottle, there was a picture of it, cleaning a toilet or whatever . . .
and i was like, wait a second, what if that bottle pictured on this bottle had a picture of itself on it? and that picture had another picture of the picture of this bottle on it and that had . . .

the concept was shocking to my juvenile mind, of course. i became depressed over the following days because i couldn't find out what would happen if that continued forever.

but anyway. now i know that NO ONE KNOWS! and that's cool. for now.
so, infinity is the size of no limit. no bounds. if something were to progress forever and ever, we would call it infinite. as in, having a quality of being endless.

take for instance, the set of natural numbers. i.e 1,2,3,4 . . .
no matter how far you go with that, there will still be more to find. no matter how big. whether it is the number of stars in the known universe,(10²³) or the number of atoms in the universe, (10⁸⁷) or even the LIFESPAN of the universe.(10¹⁰⁰) there is simply nothing bigger than that.

another example of infinity is the set of even natural numbers. i.e 2,4,6,8 . . .

but wait a second. if we match the set of natural numbers with the set of the even natural numbers, we find that the for every term in the even number set, there are two terms from the natural numbers set. this illustrated below.



so . . . by common sense, the set of natural numbers is clearly twice longer than that of the even natural numbers. right?

wrong. . as bizzare as it seems, they are exactly the length. this is because for every natural number, there is an even number to match it.

as illustrated below.



the procedure is the same with any infinite sequence. . . or is it?
for there are some infinites which are indeed bigger than others.

infact, there are two major types of infinity.

1. countable infinity.
2. uncountable infinity.

relax. this is still a maths thread. we have not ventured into English.

as the names so aptly describe, countable infinity is one that we can count. like for instance, the set of positive integers. i.e 1, 2, 3, 4 . . . these are also the natural numbers. the set of even and odd numbers as well.

you can obviously count them.

the set of real numbers on the other hand, fall into the realm of uncountable infinity. why? well, because you can't count them. literally. for instance, let us try counting the number of real numbers between zero and one.

0, 0.1, 0.2, . . . wait a minute, that's not correct at all, 0, 0.01, 0.002 . . .arrgh, that isn't correct either. . 0, 0.0001, 0.0002 . . .

eventually, you will realize that it is pretty much impossible. after 0, what comes next? what ever you think it is, you can always add another zero. forever.

they are obviously bigger than countable infinities. in fact, there are more real numbers between zero and one than there are natural numbers in their entirety.

proof?

well, let's start by matching every natural number with every number between 0 and 0.1
match 3 with 0.0 . . . 03 50 with 0.0 . . . 50 and any whole number n with 0.0 . . . 0n

after doing this, change the first number after the decimal point and you will end up with a new term which has not been paired! cool, right?

as you have probably guessed, infinity is a wierdo. . but now, it's time to see it's wierdness in full glory. below is a list of some of the amazing phenomena that infinity holds.

---

1. Hilbert's paradox of the Grand Hotel.

so, this asks, is it possible to fill a "Grand Hotel" which has an infinite number of rooms, with an infinite number of guests? well, i for one do not know. let's say you can. what then happens when a new guest arrives? there an infinite number of rooms nah, so obviously, the guest will find a room. but if he finds a room, then the hotel was never filled in the first place. but it is said there are an infinite number of guests. so, why can't they fill it?

2. The omnipotent being paradox.
i apologize in advance to theists. this might hurt a bit.

this paradox simply asks, is it possible for an infinitely powerful being, such as an omnipotent God, to create something so infinitely powerful that he/she cannot carry it?
if he can do so, then he/she just made something they cannot move, therefore, they are not infinitely powerful. if they cannot do so, well, they are not omnipotent either way.

3. The library of babel.

in this library, exists every name ever given or will ever be given, every sentence ever said or that will ever be said. the library works by generating an infinite combination of the 26 letters of the english language and the "." '','' and space. it can be found Here and has about 10⁵⁰⁰⁰ characters. compare that to the number of atoms in the universe and . . .
i also found my name there. and the day i would die . . . in a possible scenario. which is . . . 2090.

4. The Hyperwebster.
This is a theoretical dictionary that contains every word that can ever be said using the english. it obviously cannot exist in real life. but consider,
if the first term is A, the second term, AA, the third term AAA, when will the A only term ever end?
after that ends, we have AB, and then AAB, AAAB and so on till we get to zzzzzzzz . . . .
well that sucks. . .
if I had a kid and i wanted to punish him for doing something wrong, i would give him an Hyperwebster and ask him to find "Teempakguy".

The banach-tarski paradox
imagine if you can make 100 naira from 50 naira. by duplicating something and having to exact copies. the banach-tarski is a theorem that states that, it is possible to duplicate a ball into two equal balls the exact same shape and size of the original without stretching or enlarging. but by rotating. the process is insanely complicated. but relies on the fact that a sphere has an infinite amount of points on it's surface, and if they are marked by the directions in which they can be reached, if the ball is then rotated, directions cancel out and the ball mathematically transforms. if the ball is separated into different directional components[it will be four balls. for up, down, right and left directions], they transform into half the ball and when they are combined, we will have two balls! awesome! but only mathematically possible. yet.

1. zeno's paradox.
ah, this old one. this simply states that if we have two points, A and B. and you wanted to go from A to B, you would have to first pass half of that distance. but damn, in order to get to that half, you need to get to HALF of the first half . . . and so on and so forth. that at the end, you shouldn't even be moving.

---

there are a lot more situations like that. which are as bizzare, of course. like convergence and and approximate values and the circle, and pi . . . but these will be enough for now. i guess.
the concept of infinity is a very fascinating concept. as i trust you have seen now. and to a mathematical mind, it is nothing less than pure beauty. . . . sorry ladies.

so i encourage you, think about it sometime. it will do your brain good.

---

lalasticlala.
Dekatron, johnydon22, robinhez, mrsphyno , queed, xploraben, bqlekan, llaykorn, dearpreye, Blesssed, xploraben, ELYMAXiimus,Freemanan, and everyone!



Waw! really amazing post,i am physics student hence like anything mathematics,i am really greatful for this post. Waiting for another post.