How to prove that infinity minus infinity equals infinity?

Philos · 2021-11-06 23:43:57

I know that infinity minus infinity is undefined because subtracting infinity from infinity can equal anything you want. But if I want it to equal infinity, what does the proof look like?

Furthermore, how can I translate the mathematical proof into a story using physical objects. Specifically, say I have an infinite number of balls. How can I illustrate that if I take an infinite number of balls from the pile, that there are still an infinite amount of balls left?

Jai Ganesh · 2021-11-07 00:54:48

Hi Philos,

Welcome to the forum!

Please see the following link here.

What is Infinity.

Also, some more information on infinity below.

Infinity is that which is boundless or endless, or something that is larger than any real or natural number. It is often denoted by the infinity symbol

.

Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the Universe is infinite is an open question.

Bob · 2021-11-07 03:37:44

hi Philos

Welcome to the forum.

This sort of thing is why mathematicians say infinity isn't a number and the usual rules of numbers don't apply.

Nevertheless, here goes:

Arrange the infinite number of balls into two rows stretching off to ... well infinity, ... I guess. Make sure every ball in one row is paired up with a ball in the other row.

Now take away every ball in one row. You had an infinite number of balls; you've taken away an infinite number of balls. What are you left with? Well there's still one row that goes on for ever. That's an infinite number, isn't it?

I'll leave you to count them though

Bob

Philos · 2021-11-07 05:00:53

Thank you for the quick responses! What if the balls were arranged into just one roll, and they could only be taken away sequentially, as they appear in the row. If this is the case, is it then impossible to take away infinite balls and end up with infinite left to go? If so, why?

Bob · 2021-11-07 20:39:47

You can do the same thing with a single line. Mark the balls as even or odd depending on their position in the line. Then just take away the even numbered ones.

Have you heard of the hotel infinity paradox? Hotel managers ought to consider converting to an infinite number of rooms so they never run out of guest space.

https://en.wikipedia.org/wiki/Hilbert%2 … rand_Hotel

Bob

Philos2 · 2021-11-09 05:13:19

Okay, say the balls are numbered, and they have to be taken in order. If we arrange the balls into two rows, one even numbered balls and the other odd numbered balls, we apparently have two rows that are both infinite in number. Now, is it possible to take all of the odd number balls without also taking all of the even number balls? That is, since we have to take the balls in order, every time we take a ball from the odd number row, we next have to take a ball from the even number row. So once we have taken all of the balls from the odd number row, it seems there is necessarily only one ball left in the even number row. Is this correct? Or is there still an infinite number of even balls left after we have taken all of the odd number balls?

Bob · 2021-11-09 21:46:44

A mathematician called Cantor did a lot of work on infinity. He introduced the idea of a countable infinity. Basically this:

If you have some infinite set (A) and also a set of the counting numbers (B) , if you can establish a way to assign one counting number to each of the numbers in the A so that very number from each is paired once, then A is said to be countable.

It is relatively easy to show that the set of positive fractions (including 'top heavy') is countable.

https://www.youtube.com/watch?v=Q6xe0IPMFeM

However the set of real numbers is not countable. To show this he came up with a proof that, no matter how you set up your 'count', some reals are always left uncounted.

https://www.youtube.com/watch?v=0HF39OWyl54

I suggest you look up Cantor to get more information about this. It will help with your understanding of these concepts.

Bob

Philos2 · 2021-11-10 15:29:33

So, if every ball can be paired once to a natural number (the counting numbers), then the set of infinite balls is countable. Right? But I'm still wondering if there is a way to take away an infinite number of balls from the pile (in order!) such that there are still some balls left over. The closest I think is the idea of arranging the balls into even and odd numbered balls, and taking away an infinite number of balls such that all of the odd ones are taken, without taking all of the even ones. If this is possible, then it seems we can conclude at least that ∞-∞=1, since after the odd row of balls is taken, there is still necessarily one even numbered ball left that has not been taken. However, I still wonder how it would be possible to take all of the odd numbered balls without taking all of the odd numbered ones as well, since there is always going to be another even numbered ball which follows the odd numbered ball.

[I am moderately familiar with cantor's arguments. I'm trying to understand the following: "consider there exist an infinite number of members in a series. If an infinite number of members of the series have occurred, then zero members are left which have not occurred." I suspect, since infinity is so strange, that there is a coherent argument that says, "If an infinite number of members of the series have occurred, then there are still some members left which have not occurred."

I'm just not quite sure what that argument should look like.

Bob · 2021-11-10 22:37:05

hi Philos2

So, if every ball can be paired once to a natural number (the counting numbers), then the set of infinite balls is countable. Right?

Correct. In fact you can leave out the word 'if' because that's what counting is for.

there is still necessarily one even numbered ball left that has not been taken

This next idea doesn't work. Let's say that N is the even number you are referring to. Then N+1 is an odd number and you have to remove that. So you can never get to a last odd number, just leaving N.

That's why my earlier suggestion was to split the balls into two packs. That way you can remove all of one pack whilst leaving all of the other. Also what the Hotel story tells us is that you can remove an infinity of balls and leave any finite number of balls too.

Let's say you want to be left with 100 balls. Count off 100 and set them aside. Then remove all the rest. You've taken an infinity of balls and you're left with 100.

"If an infinite number of members of the series have occurred, then there are still some members left which have not occurred."

This is another way of saying you can always leave a finite number of balls, which you can, as demonstrated above.

All of which shows that there are an infinite number of ways of taking an infinite number and leaving some where some is anything from zero to infinity.

So this thread could potentially go on for ever, as you propose infinitely many, different ways of taking balls whilst leaving some (as defined above). For the sake of my sanity and the storage space on the Forum server, please would you ask something different.

Bob

Philos2 · 2021-11-11 13:52:56

haha okay thanks bob. What is your venmo or bitcoin address. I'll pay you to keep answering my questions

It seems like you are saying that if you can only take the balls in order, then there is no way to take infinity and still have some left over. Is that right?

In the example where you set 100 aside, it remains the case that these 100 were taken from the pile, so there are no left that were not taken. If you split the balls into two packs and just take one, the balls were not taken in order.

How does one take infinity from the pile *in order*, and end up with some left that have not been taken? Or, is it the case that taking infinity in order makes it impossible to end up with any left that have not been taken?

Bob · 2021-11-13 21:12:11

Or, is it the case that taking infinity in order makes it impossible to end up with any left that have not been taken?

Yes, that's it exactly.

Let's number up the infinite set {1,2,3,4,5 ... n, .....}

If you insist on taking them in order then after n you have to take n+1. So there's no chance to leave some in the set. The remainder after taking the first n, is {n+1,n+2,n+3,.....} That is still an infinite set but, by your rules, we have to take n+1 next.

At every pause, the next one to take is the first of what is left. So the set is constantly shrinking from the left.

Bob

I enjoy posting on MIF so you have already 'paid' me by asking the question.

Math Is Fun Forum

#1 2021-11-06 23:43:57

How to prove that infinity minus infinity equals infinity?

#2 2021-11-07 00:54:48

Re: How to prove that infinity minus infinity equals infinity?

#3 2021-11-07 03:37:44

Re: How to prove that infinity minus infinity equals infinity?

#4 2021-11-07 05:00:53

Re: How to prove that infinity minus infinity equals infinity?

#5 2021-11-07 20:39:47

Re: How to prove that infinity minus infinity equals infinity?

#6 2021-11-09 05:13:19

Re: How to prove that infinity minus infinity equals infinity?

#7 2021-11-09 21:46:44

Re: How to prove that infinity minus infinity equals infinity?

#8 2021-11-10 15:29:33

Re: How to prove that infinity minus infinity equals infinity?

#9 2021-11-10 22:37:05

Re: How to prove that infinity minus infinity equals infinity?

#10 2021-11-11 13:52:56

Re: How to prove that infinity minus infinity equals infinity?

#11 2021-11-13 21:12:11

Re: How to prove that infinity minus infinity equals infinity?

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