
 Ricky
 Moderator
Infinity
Infinity can have some weird properties. I invite people to post examples showing that this is true.
I'll start off with something we were doing today in Advanced Calculus:
Consider the set A = {1, 1, 1, ... , 2, 2, 2, ... , 3, 3, 3 ...}
Where "..." implies for infinity.
Believe it or not, A = {1, 1, 1...}
2, 3, 4... are never included in that set!
Last edited by Ricky (20060304 11:44:50)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: Infinity
That depends if you are doing sequenctial filling or filling in parallel. Who said we have to go left to right?
igloo myrtilles fourmis
 Ricky
 Moderator
Re: Infinity
Convention.
The notation means that the first element in that set is 1. And so is the 2nd. And the third. That's how we define set notation.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 MathsIsFun
 Administrator
Re: Infinity
Ahhh, but if we could get on board a spacheship and travel really really fast ...
Having done some programming, I agree. When a program hits an infinite loop we often say "and so it never gets to there". We used to also call that a "dynamic halt"!
I was taught about not knowing how "big" infinity is, so that ∞/∞ might be 7, or maybe 2, or, or ... , so it is undefined
How about 1^{∞}?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 krassi_holmz
 Real Member
Re: Infinity
You have to notice the continuum: The set {1,2,3,...} is countable: 1,2,3,4... The set {...,2,1,0,1,2,...} is countable too: 0,1,1,2,2,3,3... Now for tha set of all rational numbrs: write it like this: 1/1;1/2;1/3... 2/1;2/2;2/3... 3/1;3/2;3/3... .................... and count it: 1/1;1/2;2/1;3/1;2/2;1/3;1/4;2/3;3/2.... And what happens if we want to count all real numbers? WE CAN'T!!! (but if we want to count only the radicals of the positive integers, we will be able to do it) If one set is countable, it has property aleph_0. The continuun question is: Is the property of the real number set is aleph_1? for better understanding, here's a link: Wikipedia
Last edited by krassi_holmz (20060304 19:06:52)
IPBLE: Increasing Performance By Lowering Expectations.
 MathsIsFun
 Administrator
Re: Infinity
Here is one I thought of.
Is a pyramid with infinite height a prism?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 Ricky
 Moderator
Re: Infinity
I got another one like that, MathIsFun.
A striaght line is just a circle with an infinite radius.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 George,Y
 Super Member
Re: Infinity
Cantor created real numbers, and real numbers have these kinda "original sin" my belief:D
X'(yXβ)=0
 Ninja 101
 Power Member
Re: Infinity
infinity goes on and on and on and on and on and on and on.......
Chaos is found in greatest abundance wherever order is being saught. It always defeats order, because it is better organized.
 MathsIsFun
 Administrator
Re: Infinity
So, do you think we should change how we teach Infinity?
"Infinity goes on and on and on ..." is the normal idea. But it makes people think of infinity as a "dynamic" thing, and they want to see what is happening at the "unfolding edge".
But isn't Infinity the idea of where you will end up after having gone "on and on"?
Thoughts please , as I may design a page about Infinity
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 Ricky
 Moderator
Re: Infinity
I think a good idea for infinity is a number that you will never reach. But just because we can't reach it, doesn't mean we can't say what would happen if we did.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 George,Y
 Super Member
Re: Infinity
"Infinity goes on and on and on ..." is the normal idea. But it makes people think of infinity as a "dynamic" thing, and they want to see what is happening at the "unfolding edge".
Well, this defination was invented by Cauchy,A.L. and improved by Weierstrass,A.a
You have a very good question, Cauchy's defination has a flaw: Given the limit value, the variable can be as close to it as you wantbut where do this limit value come from? do you predict it? or do you simply see it in a graph?
Weierstrass, Meray,C, Dedkind,R and Cantor,G gave their solutions in between 1850 and 1900, they claim that according to Cantor's and Dedkind's defination, real numbers are continuous, hence limit manuplition can be applied to real world. Here real numbers are no longer simply rationals plus irrationals.
Their flaw was attacked by Russel, the famous British mathematician, logician and poet. He attacked that reals are dummies of rationals, and proved circular logic of barbar's(representing rationals) paradox. During half a century since then, mathematicians loving calculus and those loving logic had been quarraling about validity of reals. Logicians also took part, including Charles Sanders Peirce...
In the last years of Cantor, he proved a point equals to the whole universe according to his defination(Dedkind's defination was homogenous). He can only refer to God to defend his proposition and argue with challengers. He finally died with mental disorder.
David Hilbert is the one who ended this quarral, he proved Cantor's defination homogenous to the ancient Greeks geographic propositon a line is consist of numerous neighboring non space points. They are both true or both false.
It's upon a mathematician to freely chose to believe or disbelieve these propositions. They are the begining and cannot be proven, though with logic flaws.
Obviously, recent prof in universities won't persue logic, and according to their favor, they'd rather believe.
X'(yXβ)=0
 George,Y
 Super Member
Re: Infinity
hope you won't be landing a rocket
X'(yXβ)=0
 Ricky
 Moderator
Re: Infinity
Here real numbers are no longer simply rationals plus irrationals.
So what other type of number is in the set of reals?
Or an even better question. The normal defintion of irrational (in my experience) is not rational. So are you saying something can be both rational and irrational?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 George,Y
 Super Member
Re: Infinity
Sorry i cannot answer your question in detail, recommand you to read a math history book. Ealier mathematicians are no less smarter than us, they must had found some controversy to give a similarity to rationals and irrs, and to abandon the simple Ancient Greek grouping of ras and irras.
X'(yXβ)=0
 George,Y
 Super Member
Re: Infinity
I mean, they would not bother
X'(yXβ)=0
 Ninja 101
 Power Member
Re: Infinity
Chaos is found in greatest abundance wherever order is being saught. It always defeats order, because it is better organized.
 mikau
 Super Member
Re: Infinity
Did anyone hear Larry's comment on numbers last night?
"Ya know Aristotle said infinity is the lack of limitation. Which I suppose...is a definition of evil."
EEK! CREEEPYYYY!!! I knew calculus was evil!
A logarithm is just a misspelled algorithm.
 MathsIsFun
 Administrator
Re: Infinity
Ummm ... where is the logic in that? Could also be the definition of God. Or the endpoint of that circle that Ninja mentioned (watch out for those vicious circles!)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman
 ganesh
 Moderator
Re: Infinity
krassi_holmz wrote:Is the property of the real number set is aleph_1? for better understanding, here's a link: Wikipedia
is the cardinality of the set of all countable ordinal numbers, called ω1 or Ω, which set is uncountable, the second smallest infinite cardinal number. Infinity isn't that big That is a nice link, thanks, krassi_holmz!
Character is who you are when no one is looking.
 mikau
 Super Member
Re: Infinity
MathsIsFun wrote:Ummm ... where is the logic in that? Could also be the definition of God. Or the endpoint of that circle that Ninja mentioned (watch out for those vicious circles!)
Well he said "a" definition of evil. Not "the" definition of evil.
A logarithm is just a misspelled algorithm.
Re: Infinity
Why does an infinite pyramid have a top (pointy top), but has no base that one can imagine? The base is really big, and bigger than that, but don't forget , fully grown. I like the fully grown idea. It's interesting that we can imagine more than one infinite set, coexisting. Good thing they are fully grown, or one infinite set would get in the way of the other one. For example, the real numbers between 1 and 2, and the real 3D coordinates between (1.5 +/ 0.5, 1.5 +/ 0.5, 1.5 +/ 0.5), which is a cube with sides equal to length 1 and all the points inside it. Neat how you can have an infinite set that is a subset of another infinite set. However, the different infinities are all the same size: just infinity. Pretty neat, huh?
igloo myrtilles fourmis
Re: Infinity
to me it seems mroe symbolic than mathematical, and ironically, if something is infinite, then it is beyond our understanding anyway, and so there is no point ever trying to consider what it is like because we will always fall short
P.S i don't like thinking about infinite, too many people have spent all their lives thinking aobut it, and though it is everything, it is nothing at the same time.
Re: Infinity
Ricky wrote:Infinity can have some weird properties. I invite people to post examples showing that this is true.
I'll start off with something we were doing today in Advanced Calculus:
Consider the set A = {1, 1, 1, ... , 2, 2, 2, ... , 3, 3, 3 ...}
Where "..." implies for infinity.
Believe it or not, A = {1, 1, 1...}
2, 3, 4... are never included in that set!
... what?
Maths is like a big oven, a really big oven.
