Dtyjlui

First of all, regardless of age, people need to understand that "infinity" is not a number, and not a placeholder for a number, but an attribute of them (i.e. the fact that you can increase numbers without ever getting to an end).

For my children, the concept somehow came into their mind all alone due to the book "Guess How Much I Love You" by Sam McBratney. It plays with ever bigger numbers, and the kids can easily increase the distances used in the book on their own as soon as they learn that, for example, the sun is farther away than the moon, that stars are even farther, etc.. They had to increas their number, because otherwise I would increase mine, and "win" the contest laid out in the book.

At some age (I cannot remember if it was 5 or older), the kids figured out that they can make numbers ever larger - even larger than in the book - by adding or multiplying (doubling as in "there and back again"), or whatever operation they learned in school.

At that point, the concept of infinity seems to be represented by - literally - "in-finite", or "non-ending". I.e., they understand that there is no end to numbers, that you can go on and on adding ever more of them.

I'm not sure why the two basic things adults seem to say about infinity are "infinity is not a number" and "∞+1=∞", both of which are at best misleading. (Infinity doesn't name a number, but it does refer to a property some numbers can have. ∞+1 is nonsense, $aleph_0+1=aleph_0$, and $omega+1neqomega$.)

The problem with talking about infinity with small children is that it's an imprecise term that covers multiple precise ideas that behave differently. Children that age aren't ready for that, so I don't think there are any particular concepts about infinity that it's useful to convey. I certainly don't think there's any reason to prioritize explaining the non-fact that ∞+1=∞ over other non-facts (say, that ∞+1>∞). Rather, I think the goal has to be to enjoy playing around with the concepts.

Your son's question - how old you'll be when he turns infinity - is a really good question, and points to the difficulty of dealing with the question: it's not at all clear what the right number system to discuss that in is.

On a piece of paper, he started with writing 10, then 100, then 1000, .... and he stopped after writing 40 zeros with 1. Then he came to me and said, "I understand infinity now; infinity is a number with infinite zeros."

The main point is that as most of you suggested, he has now registered infinity in his brain as a concept rather than a number, which is why he used the expression 'a number with infinite zeros'.

Speaking as someone who was that kid, you might be able to explain $infty + 1 = infty$ via the Hilbert hotel.

Imagine a hotel that has an infinite number of rooms, one for every number. Imagine the hotel's full, and another guest shows up. You can make room for that guest by having the guest in room 1 move to room 2, the guest in room 2 move to room 3, and so on. So even though it's full, you can always fit more guests in, so you can always add one.

He's also not totally wrong to insist that $infty + 1 neq infty$. Infinity isn't really number, but an idea, that you can apply in different ways. There are different types of infinite numbers, with different rules, and in some of them, he'd be quite right that $infty + 1 neq infty$. The hotel doesn't have a room $infty$, but if it did, the room next to it would be room $infty + 1$, and would be a different room.

Five might be a bit young to understand the idea that different rules lead to different maths, but he can probably grasp the idea the idea that there are different types of infinite numbers, and it's good to tell him that he's not wrong.

My son, also 6 yo, regularly talks about millions and billions and infinity. Obviously, large numbers have some attraction to children of this age.

I try to explain that infinity is not a number. Instead, infinity is an order of magnitude which has its own algebraic rules. Plus, minus, divison and multiplication do not work the way children learn in elementary school when applied to infinity.

My first explanation is that this has also an impact on how we use the words infinity and infinite:

Three meters. (works)

*Infinity meters. (completely wrong)

*Infinite meters. (sounds wrong)

Infinitely many meters. (works)

Another approach is that the concept of numbers does not work. Numbers grow. For every number, there is another number that is larger. The mathematical notation to this concept is $forall n in mathbb{N}: exists m in mathbb{N}: m > n$. If infinity was a number, then this statement would be false, because let $n=infty$, then $infty + m > infty$ is false for all $m in mathbb{N}$ where $m > 0$. Surprisingly, children who already learnt addition upto, say, 100, understand this. They understand that 100 is not the largest of all numbers, neither is 1000, neither is a million, and so on. But the fact, that addition does not alter the "number", makes them understand that infinity is not a number.

In words that are better suited for children, you can also say: Adding any number to infinity does not change its size, because infinity expresses a magnitude, a size, rather than a number.

Admittedly, my daughter, 8 yo, understands this point better, because my 6 yo son has not yet learnt addition of numbers larger than his 10 fingers provide.

I am going to answer your question by suggesting a couple of books which might be fun to read with your son:

• 
  

  The Phantom Tollbooth by Norton Juster.

  
  
  

  The book is a rather surreal adventure trip through a Wonderland-style setting populated by mad grammarians and mathemagical wizards (among others). There is a section somewhere in the middle where the protagonist is sent on a quest to infinity, which is described in a couple of ways. The concept of infinity discussed here is, if I recall correctly, more akin to the infinity that occurs when you compactify the the real numbers, so it isn't quite the same idea as $infty + 1 = infty$.

  
  
  

  In any event, the book is quite a lot of fun. Even if you aren't overly concerned with infinity, it is worth reading, and should be at about the level of a 5 year old (maybe a little advanced? I seem to recall having it read to me when I was in first or second grade, so maybe just a little older?).

  
  


• 
  

  The Cat in Numberland by Ivar Ekeland.

  
  
  

  This book might be a little advanced for a 5 year old, but maybe not–I think that it is intended for 3rd or 4th graders, but might be accessible with the help of a parent. In any event, the book deals with infinity as a cardinal. There is the basic "a new number arrives, everyone moves up a room" example, but my recollection is that the correspondence between rational numbers and the natural numbers is discussed, as well. I had difficulty getting a copy of the book several years ago, but it appears that it might be back in print(?).

  
  

There is a well-known Christian hymn, Amazing Grace, whose last lyric captures the idea of (countable) infinity quite well, and may be more effective to a five year old because it includes a context in which the notion of infinity can be applied.

The lyrics goes:

When we’ve been there ten thousand years,

Bright shining as the sun,

We’ve no less days to sing God’s praise

Than when we'd first begun.

Obviously, this brings issues of religion into the matter. If you don't want a Christian hymn, you might rephrase the lyric in the framework of another religion or atheistically. If the child can grasp the notion of an infinite number of future years, then this lyric is "explaining" that $10,000 + infty = infty$.

I would start by saying something along the following lines...

"You're asking some very grown-up questions for someone that's only 5. Are you ready to do some really, really, grown-up thinking about the answers?"

He will of course, answer, 'yes'. I would respond...

'Ok, but this is serious stuff. You need to be ready to take this thinking very seriously.' Only when you really have his attention, do all the 'infinity is not a number' stuff, and all the other things that people have suggested.

A 5-year old, thinking about infinity, if he's really thinking about infinity, is a very smart kid indeed. He is probably smart enough to confront the idea of types of thinking beyond what he's encountered.

But he doesn't yet realise how he needs to change gear. Teach him that, before you teach him about infinity, and 20 years from now, he'll thank you for all the other things that enabled him to do.

I suppose one problem is that your son looks at $infty$ the same way he looks at $10$. But infinity is not a natural or real number, even though it has a symbol and can be used in "equations" like $infty + 1 = infty$. These equations do not have the same meaning and do not follow the same rules as with "normal" numbers — the reason being that at least one operand isn't one. Making clear that infinity is not a single number but is used to describe the unboundedness of number sequences should go a long way towards understanding and is totally within reach to a 5 year old.

Let me indulge in my computer science perspective. (I suppose it is correct in purely mathematical ways as well. Please correct me if that is not so.) How do we use "infinity" in math? For example we say that the result of some sum is infinite: $sum_{i=1}^infty{f(i)} = infty$ for some $f$. Similarly for some integral, or simply the value of some function $f(x)$ when $x$ approaches a certain value. The essence is that we make statements of the outcome of procedures. $infty$ is not a static "value"; it is a statement of what happens to a value which is computed procedurally, provided we never stop. Specifically, it is the statement that we cannot name a limit that this value will not exceed. In other words, infinity can be considered the opposite of a static, fixed value.

Making this crucial distinction will go a long way explaining why $infty + 1 = infty$ holds (even if I don't like writing it at all, as mentioned in a comment elsewhere in this thread): If I cannot put an upper threshold to a result, incrementing the result will still not yield an upper threshold, and that's all that $infty$ means.

My children both learned about infinity at around four to five years old (now 5 and 7). For both of them it was fairly straightforward; it came about with my eldest when he was talking to other kids at school about the biggest number. We talked about trillion, quadrillion, etc.; as they were at a Montessori, it was easy to understand these. Then we talked about googol, and googolplex. Then we talked about a few other numbers, like Graham's number, and the idea of other extremely large numbers.

Finally, we talked about infinity. Because we were talking in the context of very large numbers, the first thing we learned - before anything else - was that infinity is a concept, not a number. A way of thinking about extremely, impossibly large numbers, without actually naming one. This brought a little confusion, until we went through the thought exercise of: "What's the largest number. Okay, add one to it." But ultimately they got the idea of 'concept' pretty easily.

Now, at 5 (almost 6) and 7, they get some of the other ideas pretty easily - like 1/0 approaches infinity, infinty/n is infinity, but 0/0 = undefined. Teaching infinity as a concept made it easy for them to understand these are basically just rules to follow, and that it's not the same as a number.

Of course, I had a lot of sympathy that week for the poor preschool teacher who had to deal with the arguments 'infinty is the biggest number' 'no, it's a concept, not a number' between my children and the other children...

I've no idea whether this would work, but would relating it to forever hekp? Infinity is like forever but for nunbers. Doing something for a week and then forever is the same as just doing it forever. When is forever? That's when you'd be "infinity years old", but there isn't a when because forever means "never stop" . . .

So if you're travelling to infinity, which is travelling forever, when do you get there? When you stop. When do you stop? Never! Because, forever says never stop. But maybe prepare for questions about whether infinity really exists . . .

I think forever a much more familiar concept to a five-year-old than infinity, so maybe you can start from that.

Before trying to explain $infty + 1$, it'll help if he has an intuitive grasp of what infinity means. I recommend using the cardinals rather than the ordinals, because they can be constructed without needing to understand limits.

Below is a possible explanation that may help with that; I try to avoid using too much terminology, since he's 5, and in particular, when I say "number", I mean "non-negative integer" and when I say "infinity", I mean $aleph_0$. You might mention briefly that when you say "numbers" here, you mean numbers like "3", but not numbers like "3 and a half".

Start with a set with just one number in it: ${1}$. We then add numbers to it one at a time, and we'll do it so that that the biggest number in the set is also how many numbers are in the set. So right now, there's $1$ number in the set, and $1$ is the biggest number in the set. Next, we'll add the number that's one bigger than the biggest number in the set. $1+1=2$, so now our set is ${1,2}$ and the biggest number in it is $2$. We continue as such. $2+1=3$, $3+1=4$, and so on. So for any number, we have the corresponding set containing every number from 1 to that one, and our number is how many numbers are in that set.

No matter how many numbers we add, there's always a biggest number in the set and that number is always how many numbers there are in the set. But there's also always a number that's one bigger than that one. So we can keep adding numbers to the set forever and never get all of them. There is no biggest number, and there are an unlimited number of numbers; no matter how many we have, we can always add one more. So if we want to ask "how many numbers are there altogether", we need a word for that that isn't a number. That's what infinity is.

So $infty + 1$ doesn't mean anything, because $infty$ just means "unlimited"; it's specifically the thing we can't get to by adding 1 to numbers we already have.

There are a few other kinds of infinity, but this is the easiest one to understand, so you want to make sure you really understand this kind before moving on to the others.

Have you considered geometry? Take two points and draw lines, one through each points.

• Two lines meeting at an obtuse angle meet "soon," that is a small number.


• Two lines meeting at an acute angle meet "far away," that is a large number.


• Two parallel lines meet "infinitely far away."

In a way that gets you to limits, which may be too complicated.

Note that this is an Alexandroff Extension or Riemann Sphere, depending on how you look at it. With slightly older students, the Riemann sphere is a good way to "visualize" infinity.

The thing is, saying that infinity + any number equals infinity is a bit imprecice.

There are two widely used concepts of infinity; they refer to cardinality and ordinality.

In finite numbers, Cardinality is the concept of "how many" of something there are -- 1 sheep, 2 sheep, 3 sheep. Ordinality is "what order" they come in -- 1st sheep, 2nd sheep, 3rd sheep.

With finite numbers they are highly tied to each other. You can just "count labels" in a sense.

With infinite sets the two concepts diverge.

We'll start with the natural numbers -- the set of all counting numbers. 0, 1, 2, 3 etc. That'll be our "first infinity".

If you take the first infinity, and add another element to it, you get the same cardinality. This is what people talk about when they say "infinity+1 equals infinity". More than that, if you take the first infinity, and double it, you get ... the same cardinality. You can even add an infinite number of infinities to it -- take "first infinity times first infinity" or "first infinity squared" -- and you get the same cardinality.

It isn't until you reach (assuming continuum hypothesis) 2^"first infinity" that you reach a new cardinal. This can either be described as the "set of all the first infinity" or "the set of functions that go from the first infinity to yes/no" (it shouldn't be hard to see they describe the same thing). This is a bigger cardinal than the first infinity, and is also the same cardinality as the real numbers.

The cardinality game continues from there.

So that is one branch.

The other is ordinals. In ordinals, we talk about ordering things. For any two things, you can say which is in front of the other in the order. And for any collection of things ordered, we can find the "least" element (the one "behind" all the others), including the entire collection of ordered elements (we normally call this element 0).

The "first infinity" in ordinals is ordering everything by the natural numbers. Everyone gets a tag that says "1st" or "1 million and 7th" or whatever.

Now, in ordinals, we can they have someone with the label "1st in 2nd lineup", and we can state that the 2nd lineup goes after every value in the first lineup. This is "infinity plus 1" in ordinals, and it is a distinctly different way of ordering people.

What more you can have 2 infinite lineups (where the 2nd goes after the first), or 3, or an infinite number of infinite lineups (where each lineup goes after the one before). These are all distinct ordinals -- they describe fundamentally different ways of ordering things.

And the game continues from there.

Now that I have disabused you of the notion that infinity+1 always equals infinity, how do we talk about it with a 5 year old?

You could talk about that split. Say "up to infinity the idea of ordering and counting is the same. At infinity they are different."

Then talk about infinite ordering and lineups.

The ordinal $omega$ is a lineup that goes on forever. There is someone in front.

The ordinal $omega + 1$ is two lineups. One that goes on forever, and one with a single person in it. That single person goes after the first lineup. It will get very boring for them.

The ordinal $omega +2$ has 2 people in the second lineup.

The ordinal $omega + omega = 2 omega$ has two infinitely long lineups. The second goes after the first.

The ordinal $k omega$ has $k$ lineups, each infinitely long.

The ordinal $omega omega = omega^2$ has an infinite number of lineups, each infinitely long.

The ordinal $omega^2 + 1$ has an infinite number of lineups, each infinitely long, plus one person who gets to go after everyone else is done.

The ordinal $omega^2 + omega$ has an infinite number of lineups, each infinitely long, plus another lineup that goes after the previous infinite set of lineups are all done.

The ordinal $2omega^2$ has a two collections, each with an infinite number of lineups, each infinitely long, with one going after the other.

The ordinal $omega omega^2 = omega^3$ has an infinite number of collections, each with an infinite number of lineups, each infinitely long.

The ordinal $omega^omega$ has an infinitely long order of layers. In each layer, there is an infinite number of the next layer, all ordered.

You probably will break down before getting this far.

Also talk about cardinality. Here the other answers cover things really well -- things like the Hilbert Hotel and the Cat in Numberland are great resources.

A fun part of this is that every Cardinality has a whole bunch of Ordinalities associated with it. You can look at the stories, like Hilbert Hotel, and talk about how the Ordinality changed even when the Cardinality didn't.

And you can talk about how this doesn't work for "normal" numbers. You cannot change the fundamental ordinality without changing the cardinality.

Having two lines, one of 2 people, followed by a line of 3 people, is the ordinal 2+3, which is fundamentally the same as the ordinal 5. You can "just paste" the 3 people onto the end of the first line.

With infinite ordinalities, you cannot reach the end of the first line to paste the second line on. It is infinitely far away.

If you are 24 years older than him now, the answer can be "I will be 24 years older than you."

Adjust the "24" to fit.

Imagine you have a snack machine that contains infinite number of candies and gives one to you (for free) every time you press the button. So you have infinite number of candies.

Now your friend gives to you a candy of the same kind as a snack machine. Do you have more candies than you had before that? No as instead of your friend's candy you could just press a button and get it from there. The machine will continue to give them to you infinitly.

So infinity plus one is still infinity - no profit in extending infinite resource by the same thing from a finite source.

From what i know, infinity is a fun "number".

Let's say it takes one minute to climb on a rock.
If you have a tower made of 100 rocks, it takes you 100 minutes to climb on top of it.

Now let's say you have a tower made of an infinite number of rocks. It's not just big, it goes higher that the sky, it goes higher than the sun, it never stops going higher than everything that isn't infinite too.

If you try to climb on top of it, you'll never achieve to, you'll keep climbing forever, and won't reach the top even after an infinite amount of time.

If you possess a tower made of an infinity of rocks, and someone gives you another rock, you are now possessing an infinity of rocks from the tower, plus the rock you just have been given. You possess infinity +1 rocks.

If you manage to put this new acquiered rock in the tower, it still takes as much time as before to reach its top by climbing on it: forever. The tower still is made of an infinite amount of rocks.

If you possess two towers, each made of an infinite amount of rocks, you possess two infinities of rocks. But if you somehow manage to put the two towers one "on top" of the other, it takes the same amount of time to reach the top of it as it is taking to climb on the former tower.

Now let's say you have an infinitely big hole, it's so big that you can put as many rock as you want in it, it will never get full.

Even if you put your infinitely big tower in it, there will still be room left for an infinite amount of rocks. Even if you put an infinite amount of infinite towers in it, it won't change that, there is still an infinite amount of room left for rocks.

If you share your infinite amount of rocks with poeple around you, you can give them rocks forever, they all get an infinite amount of rocks. Even if you share it with an infinite amount of poeple, you'll never stop giving them rocks.

If you build a square floor made of rocks, by putting an infinite straight line of rocks in front of you, and an infinite straight line of rocks aside every rock composing the first line, it's area will be infinity².

You can then take the rocks of your floor and build a tower with it, you will never stop building the tower, which means it would create an infinitely big tower. But the tower will take much more space than the floor, because it has
an infinite amount of floors which areas are also infinity².

I don't think explaining the $infty + 1 = infty$ thing directly is very productive. It's meaning is pretty fuzzy, it is very counter-intuitive, and it doesn't give any new perspective by itself. If I wanted to explain how equality works different on infinities, I would rather try to show an example of how 2 infinities can appear both equal and unequal, at the same time. This might be e.g.:

• (natural) numbers vs even (natural) numbers. There is obviously the same amount of both, as every even number is just $2n$ for some $n$. But equally obvious facts is, that there are twice as many natural numbers as even numbers, because there is $n+1$ for each of them.




• (natural) numbers vs $n+5$ for each of them. Similar logic to the one above: there is obvious 1:1 relation between them, but also the first set has clearly 5 elements more.

This doesn't really prove anything like $infty + 1 = infty$, but it conveys a general fact of "equality for infinities is really strange, and definitely different than for numbers".

This is similar to the hotel example from a different answer, but I believe it is much better by having clearly separated sets. Instead of changing one set in some complicated way, we have two pretty simple things, lying next to each other, that we can compare in different ways.

I'm not sure whether this is a useful answer for 5-year-old. I guess other answers are much better in this regard. However, I myself struggled with these strange infinity eqalities, and examples like above were a key to get me to understand it, so I wanted to share. I believe they are the best way to explain where the "$infty + 1$" paradox comes from.