# General question re infinities

elementbrdr
Hi,

I have a question about infinity that probably stems from lacking a rigorous understanding of infinity. My understanding is that, generally, operations on infinity result in infinity. For example 2 * infinity = infinity. I am able to accept this in the abstract. However, it gets more confusing when I look at the issue in more detail. Assume two series of integers, A: 1, 2, 3, 4, 5, 6, 7, 8... and B: 2, 4, 6, 8... . For any integer x, A will have x elements and B will have x/2 elements. What I don't get is why this relationship does not hold to infinity.

I have not background in this area, so I apologize in advance if I'm asking a simple question.

Thank you.

## Answers and Replies

Staff Emeritus
Homework Helper
There are as much elements in the set {1,2,3,4,5,6,...} as in the set {2,4,6,8,10,...}

The simple reason for this is that there is a bijection between the two sets. Indeed, you can map every element in the first set to its double.

I know it's counterintuitive at first, but it's the way it is. I can't explain you why in terms that are intuitive to you, because the result isn't intuitive. You just have to see that it is indeed true and then accept it as true. Intuition doesn't help with infinities, simply because our world is finite and everything we work with is finite.

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Homework Helper
Hi,

I have a question about infinity that probably stems from lacking a rigorous understanding of infinity. My understanding is that, generally, operations on infinity result in infinity. For example 2 * infinity = infinity. I am able to accept this in the abstract. However, it gets more confusing when I look at the issue in more detail. Assume two series of integers, A: 1, 2, 3, 4, 5, 6, 7, 8... and B: 2, 4, 6, 8... . For any integer x, A will have x elements and B will have x/2 elements. What I don't get is why this relationship does not hold to infinity.
But it does! Both sets have an infinite number of elements and, as you just said, infinity/2= infinity.

I have not background in this area, so I apologize in advance if I'm asking a simple question.

Thank you.

Gold Member
2021 Award
elementbrdr, you should also be careful about using the term "infinity" since there are in fact NUMEROUS infinities, which ARE different (in fact, as I recall there are an infinite number of them).

They are called "Aleph Null" and on up.

wish you were able to get it worked out!http://www.hergoods.info/avatar1.jpg [Broken]

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SteveL27
Hi,

I have a question about infinity that probably stems from lacking a rigorous understanding of infinity. My understanding is that, generally, operations on infinity result in infinity. For example 2 * infinity = infinity. I am able to accept this in the abstract. However, it gets more confusing when I look at the issue in more detail. Assume two series of integers, A: 1, 2, 3, 4, 5, 6, 7, 8... and B: 2, 4, 6, 8... . For any integer x, A will have x elements and B will have x/2 elements. What I don't get is why this relationship does not hold to infinity.

I have not background in this area, so I apologize in advance if I'm asking a simple question.

Thank you.

Your question is actually profound. In 1638 Galileo noted the same thing ... that the set of whole numbers could be put into one-to-one correspondence with the set of perfect square numbers ... even though the latter are clearly a proper subset of the former.

Today it's simply accepted that an infinite set can be put into one-to-one correspondence with a proper subset of itself. That doesn't remove the underlying mystery. But these days we just accept it and work with it.

The underlying issue is the idea of an infinite set in the first place. 1, 2, 3, 4, ... considered all at once, as something we call a set. Where does the set of counting numbers live? Not in the physical universe; but only in our minds. After that it's all philosophy.

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DonAntonio
Your question is actually profound. In 1638 Galileo noted the same thing ... that the set of whole numbers could be put into one-to-one correspondence with the set of perfect square numbers ... even though the latter are clearly a proper subset of the former.

Today it's simply accepted that an infinite set can be put into one-to-one correspondence with a proper subset of itself. That doesn't remove the underlying mystery. But these days we just accept it and work with it.

It is not "accepted": it is one of possible several equivalent definitions of infinite set within ZFC.

DonAntonio

SteveL27
It is not "accepted": it is one of possible several equivalent definitions of infinite set within ZFC.

Just answering the question at the level it was asked. I was addressing the OP's question about infinity. Of course you're correct technically.

Still, in what branch or area of math it is NOT accepted that an infinite set is bijectively equivalent to a proper subset of itself? Either as a property or as the defining attribute? That is in fact essential to our modern understanding of infinity.

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