Question Can one infinity be larger than another infinity?

[Truthseeker007]

There are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.

 

[IG2007]

Sorry, but I think you're mistaken here. If you can continue to subtract a value, then you're not starting at infinity and you're not ending at infinity because you can ALWAYS subtract one more. If you can continue to add a value, you're not starting at infinity and you're not ending at infinity because you can ALWAYS add one more.

Again, Infinity is a concept; not a value that can be acted upon.

-Wolf sends

Sir, Infinity is not end. Infinity does not have any end. How can I end adding or subtracting? The adding / subtracting is infinite. Because, I add infinitely.

 

[Wolfshadw]

Sir, Infinity is not end. Infinity does not have any end

You're misunderstanding my comment. The End Result cannot be infinity.

How can I end adding or subtracting?

This is the entire gist of my comment. You cannot add or subtract from infinity. It's a concept that you cannot act on.

-Wolf sends

 

[Catastrophe]

Indian Genius

You are a very clever person. Infinity has nothing to do with reality. Stop trying to add or subtract from it. Infinity is just something useful in mathematics. Leave it alone when discussing reality. That is my advice from a long life. A life dedicated to science and knowledge.
My 'hobby' has been the study from the BB or 'before the BB, through astronomy and planetary sciences to the solids we stand on - geology.
I have advices for you which will appear as needed in these discussions.
Do not worry about the mental construct - infinity - leave that alone.

Cat

 

[COLGeek]

Indian Genius

You are a very clever person. Infinity has nothing to do with reality. Stop trying to add or subtract from it. Infinity is just something useful in mathematics. Leave it alone when discussing reality. That is my advice from a long life. A life dedicated to science and knowledge.
My 'hobby' has been the study from the BB or 'before the BB, through astronomy and planetary sciences to the solids we stand on - geology.
I have advices for you which will appear as needed in these discussions.
Do not worry about the mental construct - infinity - leave that alone.

Cat

Sage advice here.

 

[David-J-Franks]

As far as I have learned Maths and Physics, Infinity is something that does not have an end or is literally innumerable. But, the question is, can one infinity be larger than another infinity.?Like, a set of natural numbers is infinite. Same is a set of negative numbers. But, I think that a set of natural numbers should be greater than a set of negative numbers. Because, a set of negative numbers always add up to something negative whereas a set of natural numbers always add up to positive.

I am quite young to say that, but still, can one infinity be greater than another infinity?

To reply exactly to your question, YES one infinity can be greater than another. In fact, there is an infinite number of different types of infinity, with each one being bigger than the next. The size of an infinite set is called its cardinality and is denoted by an aleph number - aleph 0, aleph 1, aleph 2 ........... aleph infinity. Truthseeker was the closest to this in post 2, I'm just expanding on it.

It is easy to visualise this with the following description;

Aleph 0 is the smallest type of infinity and it is just the set of all the 'integers' - 1,2,3, ........to infinity. This is called a countable infinity. The next size of infinity is Aleph 1, sometimes called the continuum or continuous infinity. This consists of the set of all 'real numbers', which, in turn, includes fractions and 'irrational numbers' such as square and cube roots and transcendental numbers such as Pi and e (the natural logarithm).

The above can be visualised by imagining an infinitely long ruler. It has divisions 1, 2,3 ...... infinity, that is aleph 0 the countable infinity. Now each division between an integer can be divided into an infinitely small number of divisions and any division you care to name can always be divided again, infinitely. There are also infinitely many possible variations after a decimal place as well. This is aleph 1 or the 'continuum' and is of far greater size, ie, higher cardinality, than aleph 0. It is also uncountable. Don't ask me what aleph 2 is, I don't know, all I know is that the aleph numbers themselves go to infinity, with each set of infinites being bigger than the next!

Simple explanations;

What is Infinity?

www.mathsisfun.com

Introduction to Sets

Forget everything you know about numbers. ... In fact, forget you even know what a number is. ... This is where mathematics starts.

www.mathsisfun.com

Advanced explanations;

Infinity - Wikipedia

en.wikipedia.org

Transfinite number - Wikipedia

en.wikipedia.org

Continuum hypothesis | Set Theory, Mathematics & Logic | Britannica

Continuum hypothesis, statement of set theory that the set of real numbers (the continuum) is in a sense as small as it can be. In 1873 the German mathematician Georg Cantor proved that the continuum is uncountable—that is, the real numbers are a larger infinity than the counting numbers—a key

www.britannica.com

Transfinite number | Cantor’s theory, cardinality, infinity | Britannica

Transfinite number, denotation of the size of an infinite collection of objects. Comparison of certain infinite collections suggests that they have different sizes even though they are all infinite. For example, the sets of integers, rational numbers, and real numbers are all infinite; but each is

www.britannica.com

Fun reading;

Hilbert's paradox of the Grand Hotel - Wikipedia

en.wikipedia.org

Pigeonhole principle - Wikipedia

en.wikipedia.org

https://en.wikipedia.org/wiki/Infinite_monkey_theorem

 

[Catastrophe]

Here is a question for you. If infinity is endless, etc., etc., we are back to the universes problem. Can you have more that one everything?

 

[IG2007]

We are literally going around in circles. Cat, we all know that the first meaning of the word "universe" is not useful today. We don't know whether the universe is finite or infinite. Everything is greater than something but maybe, something might also be greater than everything (paradoxes, oof). Well, rather than going round in paradoxes, can someone state a conclusion for the problem?

 

[Catastrophe]

IG, You have my definitive answer.

It is pointless posing meaningless questions. and splitting hairs on semantic issues.

Cat

 

[Catastrophe]

Well do I remember a discussion I had many many years ago.

It was when I received my name.

I: Tell me wise Plato, how many infinities are there and which one is the largest?

Plato: Tell me, how many infinities have you experienced?

I: Well, none.

Plato: And what great unhappiness has this caused you?4

I: Well, none.

Plato: And what great happiness will it bring you when I tell you the answer?

I: Oh immeasurable happiness, for I will know the answer which only you and I know to one of the great questions of all time.

Plato: And how do I recognize infinity when I see it?

I: Oh, it will be enormously large.

Plato: And how large is the measuring stick which you will use?

I: I don’t think I have a ruler large enough.

Plato: I cannot measure infinity, and I have thought about it for decades.

I: So what should I do?

Plato: Forget about such matters which have no meaning in reality.

I: But if I thought about it for an infinite number of years, could I not then achieve an answer?

Plato: Infinity is simply something too large to measure. And, if this is its true definition, why would one waste a lifetime thinking about it?

I: Because it I interesting.

Plato: Oh, Young Man, I shall give you the name Catastrophe, for you will waste away your life on questions which have no sane answers. These are mathematical concepts. Any number divided by zero, mathematically, is infinity. So is one million divided by zero greater than one divided by zero? And do an infinite number of infinities exist? And, if they did, how would you count them? And when you grow old, having counted only 10 to the power x trillion, how will you value such a life spent on counting impossible numbers, and still not succeeding?

Catastrophe: Maybe I am wiser than all the people who have gone before me. I will be famous as the first person to count to infinity.

Plato: . . . . . . . . . <Sigh>

 

[Zamboni]

If you can measure how big an infinity is it is not infinite. So there is no way to tell if on infinity is larger or smaller than another infinity

 

[Atlan0001]

If you can measure how big an infinity is it is not infinite. So there is no way to tell if on infinity is larger or smaller than another infinity

Oh, I think there is a way:

One basket has two pockets to it. In one pocket is one orange. In the other pocket are a hundred apples. The basket is cloned to infinity. I'd say that though there are an infinite number of oranges and an infinite number of apples in that infinite number of baskets, there are still fewer oranges than apples by a 100 apples to 1 orange.

 

[Catastrophe]

" I'd say that though there are an infinite number of oranges and an infinite number of apples in that infinite number of baskets, there are still fewer oranges than apples by a 100 apples to 1 orange."

Sorry. This comes in the same category as:
I once dreamt that I won one million on the lottery and then once ten million. I'd say that the ten million is imaginary but I'll dash out and spend the one million.
It is still fantasy.

 

[Catastrophe]

I just checked at the top.
It says Science / Forums. Don't you think we should deal with this in a scientific manner? To me, that means we should see whether we can ask questions and test the answers. - not what someone might think about extrapolating the number of oranges to an imaginary delusion that infinity has any meaning in the real world.
"Number" divided zero is what it means.

Sorry, that is my honest estimation, and I accept that you have your right to your opinion. I can only wish you good luck and hope that you spend your time on more valuable subjects. As I Do, Good luck and all Best Wishes.

Cat

 

[COLGeek]

That without bounds or measure is equal to that that is without bounds of measures. No more. No less

 

[Atlan0001]

I just checked at the top.
It says Science / Forums. Don't you think we should deal with this in a scientific manner? To me, that means we should see whether we can ask questions and test the answers. - not what someone might think about extrapolating the number of oranges to an imaginary delusion that infinity has any meaning in the real world.
"Number" divided zero is hat it means.

Sorry, that is my honest estimation, and I accept that you have your right to your opinion. I can only wish you good luck and hope that you spend your time on more valuable subjects. As I Do, Good luck and all Best Wishes.

Cat

What I did is done all the time in math and physics, high school level, at least, on up. The question of this thread is "Can one infinity be larger than another infinity?" I demonstrated that it can. What I use to do that with to illustrate my answer makes no difference. I've seen many doozies of illustration in many math and physics books (including a frog trying to jump out of a hot pan).

What did "imaginary delusion that infinity has any meaning in the real world" have to do with the core math question clearly asked in the thread's question, "Can one infinity be larger than another infinity?" And moreover, the thread question had nothing whatsoever to do with [division by zero] you drop as nonsensical answer where it doesn't belong (it has no meaning at all concerning the core [topical] sense of the question).

 

[IG2007]

1 and 100 are not infinities. Sorry.

 

[DrRaviSharma]

There is difference between innumerable and infinite.
Once you think of infinity, you are agreeing that for example no number is as large as infinity.
Then how much beyond innumerable is a logical quest not easy for even math to answer.
Further the Whole Number and Infinity are connected as quoted from Yajurvada earlier.

 

[Atlan0001]

An interesting concept negative infinity is. But not going to happen except as a play thing of mathematics. Infinitesimal would be the negative of infinite. An infinite of Big Crunch mass and density has heat and the negative Big Crunch would be Big Hole an infinitesimal of mass, density and heat. Infinite, in a real world sense would have no charge at all, so would not have negative charge. Could not be anti in any respect. Which would seem to put an end to the idea of anti-gravity. Infinite gravity can exist and infinitesimal gravity can exist, but anti-gravity apparently won't. Except, maybe, as the gravity of the infinity of the non-local Universe versus the finite of the infinity of local universes. In a trees and forest analogy, the infinite number of the trees (local centers of gravity) making up the forest versus the infinite of the forest (all the trees banked up, closed up, to the non-local, the never local, the outside rim or horizon, of the forest. The Big Crunch and the big crunch walls of its translation, the Big Hole..

So a negative infinite in a real world sense would be infinitesimal, the opposed to infinite and probably wrapping to and into it to infinity (resulting in 'flat' universe physicists talk of,. One example infinite wrapping into itself to infinity, the constant of the speed of the light which is a horizon constant that can stay with any local velocity whatsoever; and thus dictate the constancy of local physics to whatever the vertical plains of locality. A traveler traveling under a constant power, a constant of self-propulsion, of one Earth gravity would never in an eternity of time (given no great interference from outside forces) know any change in local physics.

Antimatter or negative energies in the sense of finite / infinite objectivity would seem to always apply to finite.

 

[David-J-Franks]

If you can measure how big an infinity is it is not infinite. So there is no way to tell if on infinity is larger or smaller than another infinity

You don't need to measure the whole size of the infinite set. See set theory. Rather you put the 2 sets side by side and compare elements of each set to see if you can get a 1 to 1 correspondence between them. For example. Take the set of +ve integers 1,2,3 ... infinity. This is called a countable infinity. Now strip out all even numbers to get the set 1,3,5... infinity and you might think you have a set of half the size. To compare you can map each element 1 to 1 from each set. 1 maps to 1, 2 maps to 3, 3 maps to 5 etc to infinity. There is a 1 to 1 correspondence for all the elements in each set. Therefore both sets are the same size even after stripping out 1/2 the numbers. For some sets, you can't do this. For example the set of all real numbers 1 to infinity which includes all fractions, rational and irrational numbers etc.

There is an infinite number of decimal fractions between each integer and it can be shown no matter how long you make your list of fractions you can always find a number that can,t be included in your list, in fact an infinite number of them.

So, a set that hasn't got a 1 to 1 correspondence to the set of integers is called an uncountable set and is infinitely larger than the countable infinity.

This video explain it very well;

View: https://www.youtube.com/watch?v=cgpDVOZpyrI


If that's not good enough for you, then for a formal proof see Cantors 2nd theorem;

Cantor's first set theory article - Wikipedia

en.wikipedia.org

 

[David-J-Franks]

No. One infinity cannot be larger than another.

How can there be room for more than one infinity?

YES, one infinity can be larger than another;


View: https://www.youtube.com/watch?v=cgpDVOZpyrI


If that's not good enough for you, then for a formal proof see Cantors 2nd theorem;


Cantor's first set theory article - Wikipedia

 

[David-J-Franks]

Oh, I think there is a way:

One basket has two pockets to it. In one pocket is one orange. In the other pocket are a hundred apples. The basket is cloned to infinity. I'd say that though there are an infinite number of oranges and an infinite number of apples in that infinite number of baskets, there are still fewer oranges than apples by a 100 apples to 1 orange.

I think you deserve a more respectfully reply. You had a serious go at solving the problem, at first glance, it looks plausible, it's just unfortunate that its the wrong answer.
Instead of being compared to a daydreamer winning the lottery. Anyone who knows anything about maths or the answer, and who emphasises the need to be scientific, should have gone on to provide you with such a scientific answer IMO.

Whats gone wrong is that when you clone the baskets to infinity you are multiplying the apples 100, and oranges 1, by infinity. However infinity does not obey the normal rules of algebra. If you multiply infinity by any number the answer is still infinity. So the end result is you will have infinite apples and infinite oranges, both the same infinity.

 

[David-J-Franks]

I just checked at the top.
It says Science / Forums. Don't you think we should deal with this in a scientific manner? To me, that means we should see whether we can ask questions and test the answers. - not what someone might think about extrapolating the number of oranges to an imaginary delusion that infinity has any meaning in the real world.
"Number" divided zero is what it means.

Sorry, that is my honest estimation, and I accept that you have your right to your opinion. I can only wish you good luck and hope that you spend your time on more valuable subjects. As I Do, Good luck and all Best Wishes.

Cat

Don't you think we should deal with this in a scientific manner?

Seems to me that's exactly what he was doing, thought experiments are invaluable in science. It was good reasoning, to be commended.

 

[Catastrophe]

Thought experiments can be (as in this case) like dreaming about the lottery. What if I won.

Here is a real science example for you.
Two light beams coming towards you, each at the speed of light, have a relative velocity of 1 c not 2c.

Similarly, 2 x infinity = 1 x infinity. Q. E. D.

Cat

 

[David-J-Franks]

Infinity is just a word with no real meaning for us. It is a mathematical invention which is useful (?) in mathematics but no use for anything else.
Play with the word as you will. It is a word with which little or no sense can be made (except in mathematics).

Cat

It is a mathematical invention which is useful (?)

useful (?) useful (?) I would say EXTREMELY USEFUL.

There's a branch of maths called integral calculus. This deals with splitting areas and volumes into infinitesimally small pieces and combining them to find said areas and volumes. It's a cornerstone of maths, maths is the backbone of science which is a backbone of engineering. The whole modern world is dependent on modern design and engineering, which is wholly dependent on calculus, the dividing into and summation of infinitesimal small areas or volumes.

Then there's the maths of infinite series. The basic functions in trigonometry, sine, cosine and tangent can all be expressed as infinite series. Your calculator then sums up as many of the terms of these series as is necessary for your required accuracy. You couldn't build any modern structure without these functions.

All in all, you'll find calculus and therefore infinities in every nook and cranny of modern life. Many things that have been designed will have used calculus at some stage, your car, your mobile phone, your tv, etc.