Question: Infinity A bigger than Infinity B ?

[sir.lemmiwink5]

Hello everybody,

Is it possible that there can be an infinity A that is bigger than Infinity B ? How does this work/not work because i read that infinity is without limit(on wikipedia) so does saying one thing is bigger than another mean that a limit must be set to reach this conclusion or is there something i just don't understand which somebody can explain to me.

Thanks.

 

[Evo]

Infinity B will always be larger than infinity A because B comes after A.

Joking aside, looking at your posting history, are you trolling us to see how much patience we have?

 

[sir.lemmiwink5]

No i am not, i see a lot of people asking questions on this forum. I myself have genuine questions which i do not know the answer to, i am not giving an opinion or self theory which i am attempting to argue and i am neither in support of or against what is currently known in physics its more of a situation which i am trying to learn what is commonly known in physics that i do not know. This is not a homework question either as i am not involved in any educational institutions. I am just a curious person trying to learn a thing or two one post at a time.

 

[Evo]

What exactly makes you think

so does saying one thing is bigger than another mean that a limit must be set to reach this conclusion

If I say that Fred is bigger than Tommy do you think I have to explain that Fred is not infinitely large?

Please give me a specific example of something you've read where you could not tell if one of the objects discussed was infinitely large. In the every day world we know objects we deal with are not infinite, cat, apple, car, etc...

In physics, objects have descriptions which give you an idea of their size. Obviously we can't start listing these. This is where if you don't know the size of something you're reading about, do a quick web search.

And overly broad questions will not be answered, you need to do some research and then come back and post a well defined question about a specific item you are not clear on. In other words, we love helping people that help themselves.

 

[Evo]

Another mentor suggested you read this https://www.physicsforums.com/showthread.php?t=507003

 

[AlephZero]

Joking aside...

Well, it's a perfectly reasonable question about mathematics, if not about physics. (and given my PF member name, I should know )

To get started understanding this, google for countable and uncountable sets, Hilbert's hotel, and Cantor's paradox.

 

[leroyjenkens]

There's infinitely many numbers between 0 and 1, so that would mean there has to be two times infinity numbers between 0 and 2, right? Infinity never ends, so saying there's two times infinity means you're saying at the point where infinity ends, the two times infinity keeps going until it ends again. But it never ends. I think the problem lies in treating infinity as if it's a number.

 

[D H]

There's infinitely many numbers between 0 and 1, so that would mean there has to be two times infinity numbers between 0 and 2, right?

Wrong. You can make a bijective mapping between the set of numbers between 0 and 1 and the set of numbers between 0 and 2. To within an isomorphism, these are the same set.

I think the problem lies in treating infinity as if it's a number.

There are a number of ways to treat infinity as a number. Read the link cited in post #5. Also read the follow-on threads because I think cardinality is what the original poster is asking about.

 

[Integral]

The infinite number of integers is not enough to count the real numbers. Therefore there are different sizes of infinity.

 

[sir.lemmiwink5]

What exactly makes you think

If I say that Fred is bigger than Tommy do you think I have to explain that Fred is not infinitely large?

Well both would have weight or height differences with a unit of measure to show the limit of fred/tommy's height/weight, given my basic education i can understand this. What i didn't understand is how something without limit can be measured in different sizes that determine one is bigger than the other. I am not trying to be smart with you here, I'm just one individual asking others with greater mental capacity a question so i can better understand what is still a unknown to me. In addition, i understand your statement of helping people who help themselves. From now on i will search every other place but here for an answer to my questions and if this task seems to be impossible i will leave an option to create a thread on this forum as a last resort. Thanks for the link, and thanks Alephzero, leroyjenkens, D H and integral for your efforts in helping me understand.

 

[leroyjenkens]

Wrong. You can make a bijective mapping between the set of numbers between 0 and 1 and the set of numbers between 0 and 2. To within an isomorphism, these are the same set.

Yeah, that's my point. They're both infinity.

The infinite number of integers is not enough to count the real numbers. Therefore there are different sizes of infinity.

I don't understand that. To say one infinity is larger than the other, then you're saying one has to end for the other to keep going past the end of the first one. But neither of them end.

 

[phinds]

... then you're saying one has to end for the other to keep going past the end of the first one.

No, you are misunderstanding how math works. There are TONS of explanations on the internet for the various orders of infinity. Look up "aleph null" for example, as a start.

 

[leroyjenkens]

No, you are misunderstanding how math works. There are TONS of explanations on the internet for the various orders of infinity. Look up "aleph null" for example, as a start.

I've seen things like that where they basically prove you can have infinity, and then you can have another infinity that's infinity larger than the previous infinity.
Are there ways to show that the "bigger infinity" isn't infinity larger? Or are the bigger infinities always infinitely larger? But if both infinities never end, then how can one be bigger than the other. The math I've seen just shows that you can have infinity, and then you can have an infinity larger than that, but I haven't seen an explanation of how that can happen logically. Logically, if two things never end, then you can't say one is bigger than the other.

I remember in calculus, if you had a limit where the denominator was was x^2 and the numerator was x, then as x approaches infinity, you would have 0 as the answer because the denominator would exponentially increase while the numerator increases slower. But that's only as x APPROACHES infinity. It never reaches infinity. If you already HAVE infinity, then it's already reached infinity, and comparing two different sizes of infinity doesn't make sense.

 

[phinds]

You keep getting hung up on this concept you have of "infinities never end". It's basically true, but irrelevant

Again, look up, and study, aleph null and the aleph series.

 

[BobG]

The infinite number of integers is not enough to count the real numbers. Therefore there are different sizes of infinity.

What integral said.

Or, even more concrete, are there twice as many positive integers as even positive integers?

 

[D H]

I've seen things like that where they basically prove you can have infinity, and then you can have another infinity that's infinity larger than the previous infinity.
Are there ways to show that the "bigger infinity" isn't infinity larger? Or are the bigger infinities always infinitely larger? But if both infinities never end, then how can one be bigger than the other. The math I've seen just shows that you can have infinity, and then you can have an infinity larger than that, but I haven't seen an explanation of how that can happen logically. Logically, if two things never end, then you can't say one is bigger than the other.

I remember in calculus, if you had a limit where the denominator was was x^2 and the numerator was x, then as x approaches infinity, you would have 0 as the answer because the denominator would exponentially increase while the numerator increases slower. But that's only as x APPROACHES infinity. It never reaches infinity. If you already HAVE infinity, then it's already reached infinity, and comparing two different sizes of infinity doesn't make sense.

First off, you are unknowingly using the word "infinity" in a number of different ways here. That word refers to a number of distinct but yet related concepts. The infinity in $\sum_{n=1}^\infty 9/10^n$ and in $\int_0^\infty \exp(-x^2/2)dx$ are different things. The size of an infinitely large set is yet another concept.

This thread isn't about those uses of the concept of the infinite. It's about comparing the "size" of two infinitely large sets. It's about counting.So what exactly is counting? We learned to count because our innate number sense is not that good. Suppose I am holding a bunch of beads in each hand and ask you which hand holds more beads. You can answer without thinking if I have three in one hand and four in the other. You can't do that if I have 19 in one hand, 20 in another. You have to count. We humans have been counting for at least 30,000 years.

It's quite amazing that even though we've been counting for a long time, we have only very recently come to a good understanding of what counting is. The same goes for arithmetic, algebra, and all kinds of other manipulations of numbers. It wasn't until the latter half of the 19th century that the details of these concepts were hammered out. Counting is putting a set of objects in a one to one correspondence with a finite subset of the integers that starts at one and increments by one. That hand with 19 beads -- start counting from 1, each time removing a bead from the hand. When you reach 19 you'll have removed the last bead. The same goes for the hand with 20 beads, only now you have to take one more bead away. The hand with 20 beads contains more beads than the hand with 19 because {1,2,3,...20} is a "bigger" set than is {1,2,3,...,19}.

This concept of a one to one correspondence can be extended to non-finite sets. How "big" is the set of numbers {2,3,4,...}? Subtract one from each element and you have the set {1,2,3,...}. Add one to each element of {1,2,3,...} reconstructs the original set in its entirety. Having established a one to one correspondence between these two sets, we can say that they are the same size (or better, same cardinality). The same goes for the set of even positive integers and the set of counting numbers. They can be put into one to one correspondence, so they too are sets of the same cardinality. Even the set of integers can be put into a one to one correspondence with the counting numbers. The same goes for the rationals. So far, the sets discussed are all the same "size", the same cardinality.

What about the set of the real numbers? It's easy to show that all one has to worry about is the set of reals between 0 and 1; it's easy to make a one to one correspondence between this set and the set of all the reals. What about the cardinality of the reals versus the cardinality of the integers? It's easy to map the integers to the set of reals between 0 and 1. The problem is the reverse. Cantor showed that it's not possible to make this reverse mapping. Just as {1,2,3,...,20} is a "bigger" set than {1,2,3,...,19}, the set of reals from 0 to 1 (and hence the set of all the reals) is a "bigger" set than is the set of all integers. The cardinality of the reals is greater than the cardinality of the integers.

Are there even "bigger" infinities? Sure. Just as a starter, imagine all the ways to draw a curve from from some point on the line x=0 to some point on the line x=1 such that the curve always moves forwards. In other words, the cardinality of the set of all functions that map (0,1) to the reals. It's easy to map the set of all reals from 0 to 1 to this set. f(x)=constant does it. The reverse mapping is once again impossible. The set of all curves on the plane is of a greater cardinality than is the set of all points on the plane.

An interesting question quickly popped up after Cantor showed that the reals are "bigger" than the integers. That question: Is there something intermediary in size between the integers and reals, some set that is bigger than the set of all integers but smaller than the set of all reals? That there is no intermediary is the "continuum hypothesis." The validity (or lack thereof) of this hypothesis turned out to be central to one of the deepest problems in all of mathematics: Is mathematics consistent?

 

[arildno]

If I remember correctly, it is generally true that the cardinality of the power set P of a set A is always greater than the cardinality of the set A itself.

 

[BobG]

We humans have been counting for at least 30,000 years.

Coincidentally, or not so coincidentally, that's how long ago that old people were invented.

 

[1MileCrash]

I don't understand that. To say one infinity is larger than the other, then you're saying one has to end for the other to keep going past the end of the first one. But neither of them end.

You are only considering one "type" of infinity when you say this, and any two infinities of the same "type" are of the same size. When we say that infinities can be of different sizes, it is because they are two completely different things.

Furthermore, it is not about "length." We don't say that one infinity is larger than another because it is "longer." We say that one is bigger than the other for a different reason entirely, because it is, again, a completely different animal.

To see why the "infinity" of the reals is bigger than the "infinity" of the integers, consider this: what is the next integer after 1? Now what is the next real number after 1? You can answer the first, but not the second. We say that the integers are "countable" and that the reals are "uncountable." In other words, I can "make progress" in counting the first infinity, but I can't even take one step forward in the second infinity. It is not because the second is "longer," it is because it is... "denser" if you will.

These are indeed infinities are different sizes. This is because one is countable, and another is not. They are different types of infinities, and it is reasonable to say that the reals must be a bigger infinity for the reasons above. We can generalize this like so:

If I remember correctly, it is generally true that the cardinality of the power set P of a set A is always greater than the cardinality of the set A itself.

So what aldrino says is very clearly true for a finite set, but what he says is also true for infinite sets. This is called cantor's theorem.

So if we take the natural numbers, and take the power set, we must get a larger set. Thus the infinity of P(N) is bigger than the infinity of N. But, this must mean that N and P(N) are infinities of different "types" as we mentioned before. We assign a number to these "types" called aleph numbers. The "size" of N (aleph-number) is 0. The "size" of P(N) is 1, which is the same size as R. Now if we took the power set of P(N) or R, P(P(N) or P(R), the type changes again, and now the aleph number is 2. This infinity is even larger than that of R. We could do this as many times as we wanted and we would always get an even larger infinity.BUT, the key is that your intuition is not entirely wrong, it's just that you are only comparing infinities of equal aleph-number to one another. It is true that any two infinities of the same aleph number are the same size, aleph number is the only real distinction. So for example, the natural numbers and the rational numbers are both infinities and they are the same size, even though the natural numbers are a subset of the rational numbers. They are the same size because they have the same aleph number. However, the reals have a bigger aleph number than the rationals, so it is a larger infinity.

 

[D H]

However, the reals have a bigger aleph number than the rationals, so it is a larger infinity.

The reals most decidedly are "bigger" than the rationals, but exactly what the aleph number of the reals is constitutes a rather interesting problem. This turned out to be (after the fact) one of the most interesting issues with Hilbert's program.

 

[WannabeNewton]

I've seen things like that where they basically prove you can have infinity, and then you can have another infinity that's infinity larger than the previous infinity.
Are there ways to show that the "bigger infinity" isn't infinity larger? Or are the bigger infinities always infinitely larger? But if both infinities never end, then how can one be bigger than the other. The math I've seen just shows that you can have infinity, and then you can have an infinity larger than that, but I haven't seen an explanation of how that can happen logically. Logically, if two things never end, then you can't say one is bigger than the other.

Cantor's diagonal argument showing the uncountability of the reals should clear things up for you; it's a simple and elegant proof that is very visual and should make clear in what way an uncountable set is bigger than a countable set. You are using a very primitive and incorrect notion of infinity.

 

[1MileCrash]

The reals most decidedly are "bigger" than the rationals, but exactly what the aleph number of the reals is constitutes a rather interesting problem. This turned out to be (after the fact) one of the most interesting issues with Hilbert's program.

Really? I didn't know that. I remember reading a proof that there is no aleph number smaller than 1 that is greater than 0, so I just assumed that the reals must be "one step up" since I can't fathom a type of infinity that lies "between" that of countable and uncountable.

Does this apply for P(N) as well, or not? (or in other words, can we not say that P(N) and R are of equal cardinality?)

 

[WannabeNewton]

since I can't fathom a type of infinity that lies "between" that of countable and uncountable.

You may not be able to fathom it but it is not something you can prove nor something you can disprove in ZF: http://en.wikipedia.org/wiki/Continuum_hypothesis

 

[1MileCrash]

You may not be able to fathom it but it is not something you can prove nor something you can disprove in ZF: http://en.wikipedia.org/wiki/Continuum_hypothesis

So the continuum hypothesis is essentially what I am saying, but since it is independent of ZFC, a method of proof or disproof is impossible, is that right? So it is literally a hypothesis in the purest sense.

 

[WannabeNewton]

Yes it can be shown that under ZF it is neither possible to prove nor disprove the continuum hypothesis.

 

[arildno]

I think there is a deep historical irony in that when we have to deal with infinities, we basically have "to count" in the manner in which a shepherd who does not know a single number yet can check whether all sheep have returned.

We have preserved bones thousands of years old, with etched markings on them. The technique for shepherds was to have one mark for each sheep, and when they returned, let his finger go from one etching to the next.
This technique, if I remember correctly, was still observed among shepherds in 19th century Bavaria. The shepherd could perfectly well not have a single clue he had 37 sheep at all, or any other number, he could STILL keep track of whether all had arrived.

This is nothing else than counting in the shape of bijective mapping between sheep and etchings on a bone, and is the very principle we have recourse to when dealing with infinities

 

[JPMPhysics]

First off, you are unknowingly using the word "infinity" in a number of different ways here. That word refers to a number of distinct but yet related concepts. The infinity in $\sum_{n=1}^\infty 9/10^n$ and in $\int_0^\infty \exp(-x^2/2)dx$ are different things. The size of an infinitely large set is yet another concept.

This thread isn't about those uses of the concept of the infinite. It's about comparing the "size" of two infinitely large sets. It's about counting.


So what exactly is counting? We learned to count because our innate number sense is not that good. Suppose I am holding a bunch of beads in each hand and ask you which hand holds more beads. You can answer without thinking if I have three in one hand and four in the other. You can't do that if I have 19 in one hand, 20 in another. You have to count. We humans have been counting for at least 30,000 years.

It's quite amazing that even though we've been counting for a long time, we have only very recently come to a good understanding of what counting is. The same goes for arithmetic, algebra, and all kinds of other manipulations of numbers. It wasn't until the latter half of the 19th century that the details of these concepts were hammered out. Counting is putting a set of objects in a one to one correspondence with a finite subset of the integers that starts at one and increments by one. That hand with 19 beads -- start counting from 1, each time removing a bead from the hand. When you reach 19 you'll have removed the last bead. The same goes for the hand with 20 beads, only now you have to take one more bead away. The hand with 20 beads contains more beads than the hand with 19 because {1,2,3,...20} is a "bigger" set than is {1,2,3,...,19}.

This concept of a one to one correspondence can be extended to non-finite sets. How "big" is the set of numbers {2,3,4,...}? Subtract one from each element and you have the set {1,2,3,...}. Add one to each element of {1,2,3,...} reconstructs the original set in its entirety. Having established a one to one correspondence between these two sets, we can say that they are the same size (or better, same cardinality). The same goes for the set of even positive integers and the set of counting numbers. They can be put into one to one correspondence, so they too are sets of the same cardinality. Even the set of integers can be put into a one to one correspondence with the counting numbers. The same goes for the rationals. So far, the sets discussed are all the same "size", the same cardinality.

What about the set of the real numbers? It's easy to show that all one has to worry about is the set of reals between 0 and 1; it's easy to make a one to one correspondence between this set and the set of all the reals. What about the cardinality of the reals versus the cardinality of the integers? It's easy to map the integers to the set of reals between 0 and 1. The problem is the reverse. Cantor showed that it's not possible to make this reverse mapping. Just as {1,2,3,...,20} is a "bigger" set than {1,2,3,...,19}, the set of reals from 0 to 1 (and hence the set of all the reals) is a "bigger" set than is the set of all integers. The cardinality of the reals is greater than the cardinality of the integers.

Are there even "bigger" infinities? Sure. Just as a starter, imagine all the ways to draw a curve from from some point on the line x=0 to some point on the line x=1 such that the curve always moves forwards. In other words, the cardinality of the set of all functions that map (0,1) to the reals. It's easy to map the set of all reals from 0 to 1 to this set. f(x)=constant does it. The reverse mapping is once again impossible. The set of all curves on the plane is of a greater cardinality than is the set of all points on the plane.

An interesting question quickly popped up after Cantor showed that the reals are "bigger" than the integers. That question: Is there something intermediary in size between the integers and reals, some set that is bigger than the set of all integers but smaller than the set of all reals? That there is no intermediary is the "continuum hypothesis." The validity (or lack thereof) of this hypothesis turned out to be central to one of the deepest problems in all of mathematics: Is mathematics consistent?

Great post

 

[sus4]

You cannot evaluate at infinity. What you are doing is evaluating limits as a parameter approaches infinity. This CAN, but doesn't always result in a defined value. In your reference to infinity as an amount, that is more relatable to an undefined value. In calculus, we are taught that even though the value is undefined, we can still represent characteristics surrounding that value.

In this case, infinity is more of a direction than a value. Simply put, the "size" of infinity is undefined, and no comparison can be made as to which is bigger. The exception being the case where negative values are allowed, and -infinity is a direction that contains all values less than the direction of +infinity.

The closest thing to a comparison in size is to compare the rate at which some parameter in a system approaches infinity. If two systems both approach an infinite value, but one system has an increasing rate of growth and the other has a lesser increasing rate or constant rate of growth, you can see that the difference in the two systems diverge up until the point of discontinuity.

 

[D H]

You are confusing different concepts of infinity, sus4. You are writing the concept of infinity as used in analysis. That is a different concept than cardinality. There are different infinities of different sizes. I suggest you google "countability" and "Cantor diagonalization" as starting points.

 

[sus4]

You're right, I was referring to infinity as used in analysis as a parameter. However, the set of all real numbers, while an infinite set, is also an undefined set. I don't see how comparisons can be made between undefined numbers. I'm sure you are more educated than me in math, but there comes a point when you have to ask what the math means, and I don't see how two sets, both with an undefined number of elements can be compared to any accuracy.

I'd sincerely like to hear your explanation on the comparison of undefined sets, to maybe clarify it for me.

Edit: undefined in length, not in definition

 

[D H]

You compare two infinite sets by trying to put them into a one-to-one correspondence with one another. The sets have the same cardinality if this one-to-one correspondence can be constructed. If all members of one set can be mapped to members of the other set, and that other set has members that have not yet been mapped, the first set has a lesser cardinality than does the other.

Alternatively, given two sets one can assume that the two sets do have the same cardinality, that is, assume a one-to-one correspondence exists between the two sets. Now suppose you can construct a member of the second set that is not in that one-to-one correspondence. You've just arrived at a contradiction. This means the initial assumption that the two sets have the same cardinality is false.

This is exactly how Cantor showed that the reals have a greater cardinality than do the integers. Here's how he did it.

Assume the reals between 0 and 1, exclusive, can be put into a one-to-one correspondence with the integers. That means there's some real number in (0,1) that we designate as real #1, another that we designate as real #2, and so on. Now I'll construct a real number between 0 and 1 that isn't in this list. I'll start with the tenths digit of real #1 and pick some other digit that's different from that. This is the tenths digit of my constructed number. I'll do the same with the hundredths digit of real #2, the thousandths digit of real #3, and so on. My constructed number is different from real #1 by construction, different from real #2, different from real #3, and so on. It is not in my list, and yet by construction it is a real number in (0,1). Contradiction! The reals between 0 and 1 cannot be put into a one-to-one correspondence with the integers. (On the other hand, it's easy to make a one-to-one correspondence between (0,1) and ℝ.) The reals between 0 and 1 form an uncountable set, and thus so do the reals.

 

[YYaaSSeeRR]

I think there is only one infinity :) when we are talking about NUMBERS.

@EVO:
do you always review people post-history?? if yes ,why??

 

[D H]

I think there is only one infinity :) when we are talking about NUMBERS.

Three questions:
Have you read any of the posts in this thread? That you said this suggests you did not.
Have you performed the internet searches suggested in those posts?
What do you think a NUMBER is?

 

[lendav_rott]

infinity by definition is not a scalar, therefore you can't compare "different" infinities. It's just something we use when our brain goes "does not compute".

 

[D H]

infinity by definition is not a scalar, therefore you can't compare "different" infinities.

Once again, have you read any of the posts in this thread? Have you googled the search terms?

It's just something we use when our brain goes "does not compute".

Since there are examples about that show the concept does make sense, this is just wrong.


However, countability and "does not compute" bump heads in a number of ways. Here's one: The set of all functions, as shown above is an uncountable. On the other hand, the number of computer programs is a countable set. There are functions that cannot be computed. In fact, almost all mathematical functions cannot be computed.

Another one is whether it's possible to write a computer program that tests whether any given computer program will run to completion in a finite amount of time. This is the "halting problem." One way to show that this program cannot be written is to use a diagonalization argument, the same kind of argument that was used to show that the reals are not countable.

Another way to show that this program cannot be written is to assume that it can be done. Next we'll write a little python module that calls this program so we can test whether other python functions run to completion. What does this python script print?

import halting_problem
# halting_problem.halts (func) returns True if func returns in a finite amount of time,
#                                      False otherwise.

def ask_a_cretan () :
    all_cretans_are_liars = False
    if halting_problem.halts (ask_a_cretan) :
        while True :
            all_cretans_are_liars = True
    return all_cretans_are_liars


print "Are all Cretans liars? Let me ask a Cretan.\n"
print ask_a_cretan() ? "Yes\n" : "No\n"