I don't think "Infinity + 1" is possible in math

[Alessandra]

Hello good people of the math world!

I was recently having an exchange with a mathematician about infinity-related ideas. I don't know math. But it seems to me that the statement "infinity + 1" is not possible in math. Which means that neither is "infinity + 1 = infinity".

It's a question of examining the definition of infinity. If infinity is, by definition, something that could not be larger, nothing can be added to it. Otherwise you are no longer making a logical statement. Which leads me to conclude that math cannot deal with infinity, it's beyond the framework of mathematics.

My question to you guys is: has any mathematician said the same thing already? Who was that?

Sorry if this question was not posted in the correct section, but I have no idea what math section infinity falls into.

 

[pka]

the statement "infinity + 1" is not possible in math. Which means that neither is "infinity + 1 = infinity".
It's a question of examining the definition of infinity. If infinity is, by definition, something that could not be larger, nothing can be added to it. Otherwise you are no longer making a logical statement. Which leads me to conclude that math cannot deal with infinity, it's beyond the framework of mathematics.
My question to you guys is: has any mathematician said the same thing already? Who was that?

None who can be truly claim to be a mathematician would utter such a statement.
Anyone smart enough to be a mathematician knows the difference in a noun and an adjective.
The number one is a well defined set, the word 'infinity" describes the numerosity of a set.
A set is infinite if it is not finite. Both set of natural numbers and the set of real numbers between zero and one are infinite sets. But they have different numerosties. Agreed, the statement "infinity + 1" is not possible". It is a category error.

 

[stapel]

I was recently having an exchange with a mathematician about infinity-related ideas. I don't know math. But it seems to me that the statement "infinity + 1" is not possible in math. Which means that neither is "infinity + 1 = infinity".

It's a question of examining the definition of infinity. If infinity is, by definition, something that could not be larger, nothing can be added to it.

It seems as though you are thinking that "infinity" is itself a fixed (and finite, but "the largest") value. This is not true. "Infinity" is a descriptor, a cardinality.

Think about "all the counting numbers", being {1, 2, 3, 4, 5, ...}. There are infinitely many, right? You'll never run out, right? There will always be more, no matter how far out you go, right? So "all the counting numbers" is an "infinite" set.

Now add "0" to the set, so you have {0, 1, 2, 3, 4, 5, ...}. There are, obviously, still infinitely many elements in this set, right? You'll still never run out, right? There will still always be more, no matter how far out you go, right? But, this is one element "larger" than the previous set, right? Well, sort of. This is an example of how "infinity" can be a slippery concept. And is also a good example of how "infinity + 1" can be said, in an intuitive sense, to be "equal" to "infinity".

Here's another example: Think of the set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} This is obviously an infinite set. Now think of just the even integers: {..., -6, -4, -2, 0, 2, 4, 6, ...} Obviously, this also is an infinite set; but one could arguably say that it is "half" the size of the previous set, because it has only "half" of the previous set's elements. So does \(\displaystyle \, 2\, \times\, \infty\, =\, \infty?\) Well, yes, in this case.

There are even different "kinds" of infinity. You can find many interesting articles online, such as here, here, here, and here. Have fun!

 

[ksdhart]

In addition to the articles linked by Stapel, I also recommend two books. The first is "Alex's Adventures in Numberland," by Alex Bellos. Though, if you live in the United States, it was released under the title "Here's Looking at Euclid," but the content is exactly the same. The book devotes a section of about 7-8 pages to the concept of infinity and cardinalities. The second book is "Everything and More: A Compact History of Infinity," by David Foster Wallace. This book is solely devoted to Georg Cantor and infinity, and is aimed at an audience who is educated in the ways of mathematics, whereas Bellos' book is aimed at a general audience. As one reviewer on Amazon said, "...the more you know about calculus and related subjects, the more you'll get out of [Wallace's] book." You'll probably be able to find either of those books in your local library or your school's library, if you're interested.

 

[pka]

I am reminded of the time a friend asked for reference on the early life of Jesus and was referred to Dan Brown's The Da Vinci Code

Look up these creative writers Alex Bellos & David Foster Wallace .

 

[Alessandra]

I still think it is wrong

Hi again, thank you for all the answers and explanations. I looked at the first two sites linked by stapel.

I really don't agree with what's explained in this site: http://www.cwladis.com/math100/Lecture5Sets.htm

Or in the wikipedia page for Cardinality: "there are more real numbers R than whole numbers N."

The flaw in this statement is that people are still using counting to see which set is bigger and infinity cannot be counted. So any time someone comes along and starts counting to figure out things about infinity, in the same way as they do with finite quantities, they mess up.

By definition, since whole numbers are infinite, there exists no other set that is bigger (i.e., has more elements) than whole numbers. In fact, every infinite set has the same number of elements. To say otherwise is to pretend that one of the sets you are stating to be infinite is actually finite.

stapel wrote: "It seems as though you are thinking that "infinity" is itself a fixed (and finite, but "the largest") value. This is not true. "Infinity" is a descriptor, a cardinality."

Well, it certainly seems to me that cardinality is a concept that is based on finite numbers, and on counting. So it doesn't work for infinity, just like most math doesn't work when it comes to infinity - math is in many ways limited in that respect.

The minute anyone says an infinite quantity of numbers is greater than another infinite quantity they have contradicted the most fundamental aspect of the concept of infinity - which is: infinity CANNOT ever, in any way, shape, or form, be less great than anything else. A set that has a certain quantity of numbers that is smaller than another set is simply not infinite. I just see a major contradiction there. It makes no sense.

As a result, this is what I was thinking today: one way some math operations can deal logically with infinity is if the basis for the quantities expressed is actually infinity and not finite numbers.

So:

∞ - ∞ = 0

∞ * any number greater than 1 = impossible operation, since infinity can never be greater

∞ + 0 = ∞

∞ + any positive number = impossible operation, since infinity can never be greater

∞ + ∞ = impossible operation, since infinity can never be greater

∞ * ∞ = impossible operation, since infinity can never be greater

∞ / 2 = half of infinity we can never express the result with finite numbers, because it would blow up the concept of infinity, making it finite as well. The same goes for subtracting numbers other than zero from infinity, it can't be done.

∞ / 1 = ∞

That's what I think so far.

 

[stapel]

Hi again, thank you for all the answers and explanations. I looked at the first two sites linked by stapel.

I really don't agree with what's explained in this site: http://www.cwladis.com/math100/Lecture5Sets.htm

Or in the wikipedia page for Cardinality: "there are more real numbers R than whole numbers N."

Well, you're welcome to disagree and, on the basis of whatever new definitions and rules you develop, create your own new mathematics. But the articles are not asking for opinions or votes; they're merely stating established fact.

Have fun!

 

[Dale10101]

Density

"It's a question of examining the definition of infinity. If infinity is, by definition, something that could not be larger, nothing can be added to it. Otherwise you are no longer making a logical statement. Which leads me to conclude that math cannot deal with infinity, it's beyond the framework of mathematics."

It seems to me that your "definition" of infinity is, ironically, limited, and therefore your conclusion is wrong.

Mathematicians have not throw up their hands and said, INFINITY, "it's beyond the framework of mathematics" and turned to playing their flute.

They have noted that if you assign "infinity" a symbol and try to use it as in the conventional way as an argument in the usual binary relationships, like addition, subtraction, multiplication etc, the calculation stops because this new symbol was not part of the domain used to define the operation in the first place.

What to do? Grab your flute? No, ponder the plenitude.

Meditating mathematicians noticed, for example, that all infinities are not equal, some are more “dense” then others. Compare the infinity of symbols found between 0 and 1 on the real number line with the totality of the number of symbols found on the entire number line of natural numbers. The point here is that there is a lot more to infinity than ‘beyond big’. In a rough way one might say that beyond its ‘scopelessness’ infinity has a “nature” and within that context mathematicians define properties that are useful. Calculus in particular has needed a close examination of the concept of infinity to make sure that the whole delta - epsilon type of proof (If I can prove that you cannot prove me wrong then I must be right!) doesn't fall apart.

Contemplating infinity is almost a spiritual exercise and can leave one cross-eyed, or so I have found. You might possibly be as nuts as I, good luck with your journey in the realm of mathematical imagination.

 

[Alessandra]

It seems to me that your "definition" of infinity is, ironically, limited, and therefore your conclusion is wrong.

Mathematicians have not throw up their hands and said, INFINITY, "it's beyond the framework of mathematics" and turned to playing their flute.

They have noted that if you assign "infinity" a symbol and try to use it as in the conventional way as an argument in the usual binary relationships, like addition, subtraction, multiplication etc, the calculation stops because this new symbol was not part of the domain used to define the operation in the first place.
...
Contemplating infinity is almost a spiritual exercise and can leave one cross-eyed, or so I have found. You might possibly be as nuts as I, good luck with your journey in the realm of mathematical imagination.

Hi Dale10101,

Thanks to a discussion on another math forum, I now think I have some idea why Cantor (or whoever it is) said things like "there are more real numbers R than whole numbers N." So I think I've understood the concept of comparing infinite sets. And why this is a very different operation than when you manipulate infinity as the totality of the infinite elements, like ∞ - ∞ = 0.

I have greatly enjoyed thinking about infinity from a math perspective - as limited as my understanding of math is! I have now a newfound appreciation for the natural number set, because they are so simple and elegant - and infinite!

I had another thought today related to infinity. I was thinking about what is different between the entity of a set and the entity of a numerical quantity. And I did the following thought experiment.

Take a circle. Now inside this circle, for every real number, you put a point. So how many points inside the circle? Infinite. Then you take another circle next to it. Inside this other circle, for every integer, you put a point. How many points do you have inside the second circle? Infinite. Then I tell you, now remove from each circle an infinite number of points. Can you remove an infinite number of points from the integer set and still have any points left over? I can't picture it. I'd say no. Because where do you draw the line between finite and infinite? I find it hard to picture that. What do you think?

Second question: for the real number circle, can a portion only of the points be infinite? Once they all become points, and are no longer distinct numbers, wouldn't the real number circle function as the integer circle?

I think it's fun to think about this. But I'm glad I'm not a mathematician, because as you say, I'm positive the little sanity I can ever make a claim to would go down drain, and quickly!