# Question: Infinity A bigger than Infinity B ?

## Main Question or Discussion Point

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.

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Evo
Mentor
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?

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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. Im just a curious person trying to learn a thing or two one post at a time.

Evo
Mentor
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.

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Evo
Mentor

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AlephZero
Homework Helper
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.

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

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.

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
Gold Member
2019 Award
... 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.

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
Gold Member
2019 Award
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
Homework Helper
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
Staff Emeritus
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
Homework Helper
Gold Member
Dearly Missed
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
Homework Helper
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.

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
Staff Emeritus
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.

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

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