Which infinity is bigger?

[Agent004]

Imagine you have the line of real numbers before you, now you only look at the interval [0, 1] (or any interval with the range is 1). Tell me how many real numbers exists in that interval.

Now considering the whole real number line again, tell me how many numbers existings in the real line.

Now which one is bigger, the number of real numbers from the interval [0, 1] or the number of real numbers from the entire real line

Prove it mathematically

 

[bizmark]

They're the same. A number called Aleph if I'm remembering correctly. Give me three weeks and I'll post a proof (which will be blatantly ripped off from somewhere). I just can't right now. I've got too much work to do.

 

[rimshaker]

Theoretically, the infinities are the same. They can go on forever because there is no largest or smallest number. But realistically and physically, it's being proven in quantum physics that even spacetime has a finite discrete amount of space and volume... on the Planck scale. Smallest area possible is h^2, and smallest volume is h^3. Again, this is related to the utlimate theory of quantum gravity so it's not written in stone yet.... but there has been proof that spacetime is not continuous.

Doesn't make sense to ask which infinity is bigger. Infinity is a single concept, and mathematically there is no difference between x/0 and x*infinity.

 

[Agent004]

rimshaker, I perfectly understand what you are saying. But the point I am trying to get cross is this

1) Do you agree that the whole real line is bigger than the interval [0, 1]
2) Wouldn't it imply the infinity from the whol real line is bigger than the infinity from [0, 1] <---- is this part

I too agreed that they are the same but... well let me just say that axioms are not so applicable when we are dealing with infinity.

Perhaps my maths degree haven't taught me enough yet.

Edit: typo

 

[CTho9305]

They are equal... both are the 2nd infinity, the first inifinity being the number of integers. there are more reals than integers. but there are not more integers than, say, even or odd integers. I dont remember exactly why... but probably b/c there are infinity reals between every two reals or something related to that. . the symbol used for infinities is a hebrew symbol, IIRC.

(learned that in 7th grade . I still dont really understand infinities very well, or see any particular use for them though )

edit: the symbol is in fact aleph... if you search google you'll find a lot of sites with a pic of it.

 

[AnthraX101]

Of course, this is incorect. This is why I hate Calc. Sorry, but i'll leve it for posterity.



<< The set of all real's.

First, for the logical proof: [-inf, inf] includeds [0,1] Therefore, since it is encompasing the other, and more, it would be larger.

Mathmaticly? Take the ratio [-inf, inf]/[0,1] . Next, put a sigma in front of both. Write it out. You will be able to cancel out the botom, and leave the top. Take the derivitive of it (L'Hospital's rule) and you will see it diverges.

Now flip it. [0,1]/[-inf,inf] Do the same steps. You will get 0 over infinty. Once again, L'Hospital it. You will find it converges to 0.

The top, therefore, must be smaller then the botom.

Armani >>



Armani

 

[CTho9305]

hmmm... but both are bigger than the set of integers right? i hate number theory

 

[Agent004]

<< hmmm... but both are bigger than the set of integers right? i hate number theory >>



Yes that's correct, both sets of real numbers are bigger than the set of integers

 

[Turkey]

There's a difference between saying one infinite set of numbers is larger than another infinite set of numbers, and asking which infinity is greater. Infinite is a concept that means no boundaries, therefore one infinity is not larger than the other infinity. But one set is larger than the other, ie Armani's proof.

 

[Jonitus]

Okay, wrap your brain around this.

Give me the expression for a function which is defined everywhere along the number line, yet is never continuous.

This is a good example of infintesimals, both large and small.


f(x) = 1 if x is rational
f(x) = 0 if x is irrational


between any two rational numbers, there are an infinite number of irrational numbers. Likewise, between any two irrational numbers, there are an infinite number of rational numbers.

Q.E.D.

 

[RossGr]

It is all about counting, to determine the cardinality of a set you attempt to create a one to one correspondence between the set of integers and the set in question. If the set is finite it is easy,the cardinality is the number of elements. If the set is infinte and you are able to create a one to one correspondence between each element and the set of integers then the set has the same infinity as the integers Aleph nought. This is possible for the set of fractions, which is called the rational numbers, but not possible for the real number line or any segment of the real number line. The real numbers have cardinality Aleph sub one, the cardinality of the continum. This is the cardinality of real numbers and ANY interval of the real line. Sets of higher cardinality are formed by forming super sets of the reals, a superset consists of all of the possible subsets of the real numbers.

So the intervals (0,1) or [0,1] or (-inf, inf) are cadinality Aleph sub 1.

 

[Agent004]

<< It is all about counting, to determine the cardinality of a set you attempt to create a one to one correspondence between the set of integers and the set in question. If the set is finite it is easy,the cardinality is the number of elements. If the set is infinte and you are able to create a one to one correspondence between each element and the set of integers then the set has the same infinity as the integers Aleph nought. This is possible for the set of fractions, which is called the rational numbers, but not possible for the real number line or any segment of the real number line. The real numbers have cardinality Aleph sub one, the cardinality of the continum. This is the cardinality of real numbers and ANY interval of the real line. Sets of higher cardinality are formed by forming super sets of the reals, a superset consists of all of the possible subsets of the real numbers.

So the intervals (0,1) or [0,1] or (-inf, inf) are cadinality Aleph sub 1. >>



So they are the same or equivalent to each other?

 

[RossGr]

The cardinality is the same.

 

[Agent004]

Thanks

This answers my original question (badly phrased ).

 

[bizmark]

<< there are more reals than integers. but there are not more integers than, say, even or odd integers. I dont remember exactly why... >>



The cardinality of even integers is the same as the cardinality of integers because you can define a one-one map between the two (f(x)=2x). Then every integer has its corresponding even integer, and every even integer has its corresponding integer (could still be even).

The rational numbers can be determined similarly by a one-one map from the integers. Thus they have the same cardinality.

This first step was crucial for me to gain a better conception of infinity. It seems clear to everyone that there are more integers than even integers. But there's a one-to-one correspondence between the two. That's the nature of an infinite set, even a proper subset of it can still have the same cardinality as the whole set.



<< Okay, wrap your brain around this.
Give me the expression for a function which is defined everywhere along the number line, yet is never continuous.
This is a good example of infintesimals, both large and small.
f(x) = 1 if x is rational
f(x) = 0 if x is irrational
between any two rational numbers, there are an infinite number of irrational numbers. Likewise, between any two irrational numbers, there are an infinite number of rational numbers.
Q.E.D. >>



Switch around your function s.t. f(x)={1, x irrational; 0, x rational. Then the Lebesque integral from 0 to 1 of this function is defined and is in fact 1! This is because, even though between every two irrationals there is a rational, and vice-versa, there are still many, many more irrational numbers than rational.



<< Infinite is a concept that means no boundaries, therefore one infinity is not larger than the other infinity. >>



Except in the case of aleph-null and aleph-sub-one. Both are infinity, but one is the cardinality of the integers and the other is the cardinality of the real numbers.

 

[rimshaker]

<< rimshaker, I perfectly understand what you are saying. But the point I am trying to get cross is this

1) Do you agree that the whole real line is bigger than the interval [0, 1]
2) Wouldn't it imply the infinity from the whol real line is bigger than the infinity from [0, 1] <---- is this part >>



No, it still doesn't work when you think of it that way... nice train of thought though. The real number line is infinite both ways, and the 'gap' between [0,1] is infinite in terms of decreasing value. But to say one is bigger than the other is implying there are 2 separate values to compare... and that's my point.... the idea of infinity is just a single idea or entity. One approach to infinity doesn't make it more/less important than some other approach to get infinity.

 

[Jzero]

Do some research on Georg Cantor's continuum hypothesis. It is unproven still, but his idea was, in a nutshell, that there are different "degrees" of infinity. You can come up with an infinity, aleph-null and find an infinity that is numerically larger than aleph-null (call it aleph-one).
The Infinity Hotel story is a common illustration of the concept.

Rudy Rucker once theorized:
If you had a substance "ether" and could split that substance into an infinite number (aleph-null) of point-sized (read zero) particles
You could compare it to a substance "matter" which could be split into an equal infinte number (aleph-null) ether-sized particles. You could further break those particles into a larger infinite number (aleph-one) of point-sized particles.
This doesn't hlep much since point-sized particles are too small to "count."
But, if you had a substance "bloog" which could be split into aleph-null matter-sized particles and aleph-one ether-sized particles, you would see easily that the pile of ether particles from bloog (aleph-one) was larger than the pile of ether particles from matter (aleph-null).

For a brilliant fictional exploration of the properties of infinities, take a look at Rudy Rucker's White Light--a story of a set theorist gone off his rocker.
Or check out some of Dr. Rucker's non-fiction explorations.

PS--How did I get into the HT forum? I swear this was the OT forum...

 

[Hanpan]

<< you would see easily that the pile of ether particles from bloog (aleph-one) was larger than the pile of ether particles from matter (aleph-null). >>



Correct me if I'm wrong here but two things.

Firstly your point particles should still be coutable (assuming that we use integers to count them. An infite set of intergers is countable I belive)

However I don;t really follow the size analogy. If you have an infinte number of particles, their combined size (assuming they have some size) would again be infinite. So you are back to which is larger infinity or infinity.

 

[Jzero]

<< [Firstly your point particles should still be coutable (assuming that we use integers to count them. An infite set of intergers is countable I belive) >>


Not countable in that one assumes that matter, ether and bloog are all visible to the human eye even in particulate form, but you wouldn't be able to see a "point" based on the most literal definition of one.



<< However I don;t really follow the size analogy. If you have an infinte number of particles, their combined size (assuming they have some size) would again be infinite. So you are back to which is larger infinity or infinity. >>


Quite true, which is why the aleph-null and aleph-one terminologies come into play. Both are infinite, but one contains more cardinals.

If I can break one unit of matter into aleph-null units of ether
and I can break one unit of bloog into aleph-null units of matter which can in turn each be broken down into another aleph-null units of ether, you can easily see that while both values are infinite, it is quite obvious that I was able to produce more units of ether by breaking down bloog than by breaking down matter alone.
Both piles will contain an infinity, but the pile that came from the bloog will be bigger (contains more units).

Goes back to my original statement about Georg Cantor theorizing different degrees of infinity. At the basic level, infinity is infinity, but if you abstract further, there is more than one infinity.

If you read through the link above, for instance, there is a scenario:
A hotel with an infinite number of rooms and no vacancies is visited by an infinite number of busses full of people.
The hotel contains aleph-null people.
In order to fit the new infinity of people into the hotel room, the proprieter asks everyone currently in an odd-numbered room to double their current room number and move into the corresponding evenly-numbered room.
He has now freed up an infinite number of odd-numbered rooms and places the new infinity of guests in each one.
The proprieter just stuffed an infinite number of additional people into a hotel already containing infinity.
There is clearly a larger infinite number of people now (aleph-one) than there was when we started.

 

[Hanpan]

Again I beg to differ.

Take the piles of matter. Each pile would be infinatly large, ie taking up more space than teh known universe (whihc is not inifnate.)


How then can you say that one is bigger. Bigger infers some sort of notion of space. But an infinate number of particles of any size would occpy infinate space. As for a hotel with infinite rooms, I contend you cannot fill it. However if you do use infinite people to attempt to fill it, it would take all possible people and then more. Perhaps I cannot correctly comprehend infinity however, to my understanding, an infinite hotel would never be full, for even as you add infinite people there would always be more room, for as many people as you wish to add.

I have however become courious and will consult with one of my math profs tomorrow whose area of experties is number theory.

 

[Jzero]

<< Again I beg to differ.

Take the piles of matter. Each pile would be infinatly large, ie taking up more space than teh known universe (whihc is not inifnate.) >>


When I say that one pile is "bigger" physical size is irrelevant. I'm talking about it being NUMERICALLY bigger. There are simply more pieces in the aleph-one pile than in the aleph-null pile.
Further, there's no reason to believe that a physical representation of infinity would have to exceed the universe--if I could break a rock the size of my fist into an infinite number of tiny pieces, they obviously couldn't take up any more space than the rock itself.




<< How then can you say that one is bigger. Bigger infers some sort of notion of space. But an infinate number of particles of any size would occpy infinate space. >>


Again, bigger strictly in a numerical way.
If you have a pile of 10 pieces of candy and another pile of 50 pieces of candy, you can eyeball and count by comparing each pile on a one-to-one basis. You'll run out of candy in the 10-pile and still have candy left in the 50-pile, so the 50-pile is obviously numerically greater.
Likewise, compare the bits of ether on a one-to-one basis and you will run out of particles in the pile that came from matter before you run out of the particles in the pile that came from bloog.



<< As for a hotel with infinite rooms, I contend you cannot fill it.
However if you do use infinite people to attempt to fill it, it would take all possible people and then more. Perhaps I cannot correctly comprehend infinity however, to my understanding, an infinite hotel would never be full, for even as you add infinite people there would always be more room, for as many people as you wish to add. >>


Divorce yourself from physical limits for this. Of course there will never be infinite people to fill the hotel. You can only approach it from a theoretical perspective. If you DID have an infinite number of people....

Anyway, for this example, there's no need to assume the hotel is filled, so we'll drop that part, and I'll try to make it fit better with the prior posts concerning cardinality (I don't do well with Math-speak and I found that Rucker's more pictorial explanations in White Light were much easier to comprehend).
The hotel has infinite rooms, all empty. You can put someone in any room in the hotel, but you must be able to give them a specific room number.
An infinite number of busloads of people shows up at the door. Those infinity of people (let's give them a value of aleph-null) stand in line and you give each one a room starting with room #1 and working your way up.
Eventually you have filled aleph-null rooms with aleph-null people. The line of people has dissipated. (At this point, I claim the hotel is "full" because it has an infinite number of rooms and into those rooms we placed an infinite number of people. Also, assume one guest to a room).
Suddenly, ANOTHER infinite number of people shows up!
Since aleph-null is infinite, we can't really assign a discrete value to its limit. You forgot the cardinal number of the last guest you placed. We can't tell the next guests to go to the aleph-null + N room because they will never be able to walk and reach aleph-null. We've got to find another way.
If everyone in an even numbered room doubles their room number and moves into THAT room, we will have vacated an infinite number of odd rooms and you can tell the guests to go into each odd room.
Another infinity of guests goes into the rooms. THere's a greater infinity of guests now than there was before this latest infinity showed up. So there's aleph-one guests. Still infinity, but a higher degree.


<< I have however become courious and will consult with one of my math profs tomorrow whose area of experties is number theory. >>

.

In a nutshell, what Cantor thought but could not prove was that if you could assign a numeral to each value from 1 to infinity, if you could exhaust all numerals, there would still be more values. Since there's an infinite number of numerals, the total number of values available must be larger.

But it's unproven--all that I've said is theoretical. And being that I'm not a mathematician in any way, let alone a set theorist, I could be off about a few things...it's been 4 years since I read White Light.

 

[Hanpan]

I still disagree, however I will not make any more arguments untill I have some time to dicuss this with someone who has the proper math background.

 

[GDX]

Just to clear something up. Cantor's Continuum Hypothesis was not that there are different infinities. It was that aleph_1=C (that is, that aleph_1 is the same as the cardinality of the reals). This can also be stated as, there does not exist an infinite set with a cardinal number between that of the integers (countable infinity, also know as aleph_0) and that of the reals (cardinality C, the "continuum").

Furthermore, the issue of "proving" it has become moot. It has been proved that it can't be decided from the normally accepted axioms of set theory (Zermelo-Fraenkel Set Theory).

In general however, there is a tendency to want the Continuum Hypothesis to be false, and some additional to the Zermelo-Fraenkel axioms have been proposed which make it false.

A great site for all many math questions is http://mathworld.wolfram.com
Continuum Hypothesis info can be found there at http://mathworld.wolfram.com/ContinuumHypothesis.html
General information on cardinality at http://mathworld.wolfram.com/CardinalNumber.html

~GDX

 

[wurmyhi]

side question: Was Rudy Rucker the author of "Spacetime Doughnuts?"

 

[Jzero]

GDX appears to know more about this than I

Rudy Rucker was the author of Space-Time Doughnuts and a number of other nifty tales....