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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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 (2006-03-03 12:44:50)*

"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..."

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

That depends if you are doing sequenctial filling or filling in parallel.

Who said we have to go left to right?

**igloo** **myrtilles** **fourmis**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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..."

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,687

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[sup]∞[/sup]?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

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 (2006-03-03 20:06:52)*

IPBLE: Increasing Performance By Lowering Expectations.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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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

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

I got another one like that, MathIsFun.

A striaght line is just a circle with an infinite radius.

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,333

Cantor created real numbers, and real numbers have these kinda "original sin" --my belief:D

**X'(y-Xβ)=0**

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**Ninja 101****Member**- Registered: 2006-02-20
- Posts: 936

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.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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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

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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.

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**George,Y****Member**- Registered: 2006-03-12
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"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 want--but 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'(y-Xβ)=0**

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**George,Y****Member**- Registered: 2006-03-12
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hope you won't be landing a rocket

**X'(y-Xβ)=0**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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?

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,333

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'(y-Xβ)=0**

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,333

I mean, they would not bother

**X'(y-Xβ)=0**

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**Ninja 101****Member**- Registered: 2006-02-20
- Posts: 936

infinity is a circle.

Chaos is found in greatest abundance wherever order is being saught. It always defeats order, because it is better organized.

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**mikau****Member**- Registered: 2005-08-22
- Posts: 1,504

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.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,687

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!)

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 27,026

krassi_holmz wrote:

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 Is the property of the real number set is aleph_1?

for better understanding, here's a link: Wikipedia

That is a nice link, thanks, krassi_holmz!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**mikau****Member**- Registered: 2005-08-22
- Posts: 1,504

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.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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, co-existing.

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 3-D 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**

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

Yeah John, pretty neat indeed!

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**thinkdesigns****Member**- Registered: 2005-07-21
- Posts: 23

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.

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**jimmymcjimmington****Member**- Registered: 2005-05-18
- Posts: 58

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.

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