How can I familiarize elementary school students with infinities larger than $\aleph_0$?

Question

Cantor's discovery of the existence of more than one infinity was a revolutionary change in human knowledge. He defined the notion of counting by bijections and showed that one can use infinities as numbers for counting mathematical objects.

I am planning a "Set Theory for Children" project to familiarize elementary school students with the basic notions of set theory.

There is no problem with giving an intuitive definition of infinity. Children in elementary school are already familiar with counting by natural numbers. One can convince them that the set of all natural numbers is infinite because it "never ends" and one can find a number larger than any given number by adding 1 to it.

But there are few concrete examples for infinities larger than $\aleph_0$. Also introducing basic transfinite arithmetic in a naive intuitive way can easily lead to paradoxes like Hilbert's Hotel.

Question. How can I explain the existence of infinities of different sizes and transfinite arithmetic so elementary school students will understand it?

Remark. I believe transfinite numbers and their arithmetic are as natural as finite numbers and finite arithmetic. The fact that they seem strange even for professional mathematicians has a root in our elementary school education and our incorrect/incomplete basic intuition about the notions of "number", "enumerating" and "arithmetic". So there is a natural way to teach these natural notions to students even in elementary school. We just need to discover such a teaching method.

Answer 1

I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised.

I do think that exploring counting from both an "ordinal" and a "cardinal" point of view is probably a good idea.

Example for a 5 year old: Something you could do is have 20 stuffed animals, only 18 of them wearing hats. You can ask: are there more hats or stuffed animals? Get them to explain their reasoning: How could you know there are more stuffed animals than hats without counting them? These kinds of questions introduce them to the ideas of injection, surjection, etc, at a level appropriate to the age.

Answer 2

You can easily make them draw $\aleph$s; however the rest is much more demanding. There is a nice analysis from a researcher group taking a constructivist perspective. They distinguish potential infinity from actual infinity. The first one is covered by the idea of "counting on forever" the second one needs infinitely many things to exist in your thinking at the same time. Taking the step from potential to actual infinity seems to be even more demanding than taking the step from natural numbers as ordinals to natural numbers as cardinals (both times, the view has to change from process to object). The latter step takes children (roughly) two years. Consequently, the literature they cite does not find people up to the age of 15, who manage the transition to understanding actual infinity. This might be due to teaching as you indicated.

I believe that understanding different infinities needs at least understanding $\aleph_0$ as actual infinity. So your goals seem very ambitious.

Literature

Dubinsky, E., Weller, K., Mcdonald, M. A., & Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1. Educational Studies in Mathematics, 58(3), 335–359. doi:10.1007/s10649-005-2531-z

Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos Analysis: Part 2. Educational Studies in Mathematics, 60(2), 253–266. doi:10.1007/s10649-005-0473-0

Answer 3

Consider a Sumerian person, living around 2500BC, who owns a flock. She hires shepherds to take the flock to pasture. Not being a scribe, she does not know Sumerian numbers- instead, for each sheep that passes her, she places a stone in an urn. When the shepherd returns, for each sheep that comes back, she removes a stone from the urn. If the last sheep passes and the urn still has stones in it, the shepherd has been doing something funny. If the stones run out, but there are still sheep, it's her lucky day. If stones and sheep run out at the same time, all is in order.

The Sumerian person in question can compare urns in the same way- remove one stone from each, at the same time. A finite urn can be emptied by removing stones one at a time. An infinite urn cannot.

Consider now two infinite urns. If we can match each stone in urn A to a stone in urn B, then urn B is at least as large as urn A. But if, after matching up each stone in urn A to a stone in urn B, there are still unmatched stones in urn B left over, then urn B is larger than urn A. You can visualize elements in urn B as stones and elements in urn B as Greek-style points with no volume to add conceptual intuition.

Answer 4

I would expect children can understand the idea that there are as many even numbers as there are natural numbers, as long as it's presented in a lively style, asking them questions and drawing pictures.

How many numbers are there? Infinity. (Write the first few naturals)

Can you tell me what an even number is? What are the first few? (Write them down) How many even numbers are there? Infinity!

Are there more numbers than there are even numbers? Yes, hmm, they're both infinity

But look, if I write the even ones under the ordinary numbers like this...

The trouble with higher infinities is that there are no uncountable sets that children are familiar with. Even middle school students may be uncomfortable with the idea that there are infinite decimals. The simplest example would be the set of all infinite sequences of ones and zeroes, but I suspect that would be too abstract for small children (and possibly some adults) to grasp without some difficulty. You might replace the ones and zeroes with something more colorful, ideally something that could be wrapped up in a story, but the concept of "an infinite set of infinite sets" might never stop being difficult. The diagonalisation argument is accessible for high school students and probably middle school students, but I can't imagine an 8 year old being able to maintain all the structures in their head through to the conclusion.

So if we're resigned to the idea that proving the existence of uncountable sets is too hard, can we at least communicate what their existence means? I think that to do this, you should first establish the idea that there are a lot of countably infinite sets. I'd expect kids to be surprised that there are as many even numbers as natural numbers (surprise them further by repeating the argument with multiples of a billion million trillion, rather than multiples of two). What about an infinite checkerboard (with a top-left-hand corner)? First number the first row with the natural numbers, to make it look like there are way more squares than natural numbers. Then ask if they can think of a way to number the squares so that they all get a number, and if they don't come up with anything, show them the correct answer. The crucial part is to set it up so that they're surprised that an infinite checkerboard is countable.

There might be a few other simple countable sets, and once that's done, you can ask them if they think all infinities are the same size. Then you can drop the existence of uncountables on them. I would finish with something like "these bigger infinities were first discovered by a man named Georg Cantor about a hundred and fifty years ago". I feel it's important that they understand that the fact that certain sets are uncountable is something that a person can prove, and simply giving the name of the first human to do so might be enough to communicate that, on an emotional level at least.

Answer 5

If the idea of "larger than infinity" caused a riot among grown up mathematicians, accustomed to rarefied abstract thinking for years if not decades, this will just make their little heads explode.

In my automata theory classes I have to get my students (by then mangled though school and two years of mathematics at college level) to grasp Cantor's diagonal argument, and I feel I'm unsuccessful most of the time.

(Yes, it is an extremely beautiful result. But not everyone is prepared to appreciate it as such. Tough luck, move on.)

Answer 6

Power sets, maybe? I was just pondering this because of course preschoolers don't understand decimals.

Answer 7

Depending on their age, they may not have reached what Piaget called the “abstract operational” stage yet. If they’re still in the concrete operational stage, they will need physical or otherwise concrete explanations. This is one reason why we don’t attempt algebra at 5th grade—their minds haven’t gotten there yet.

Answer 8

Los Alamos National Laboratory wrote a nice lesson plan in the mid-90s around an embellished version of the Hilbert Hotel story, including the alignment to the 1989 NCTM standards:

Hotel Infinity (you'll need to click around, as the interface is very 90s...) http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html

If you prefer a youtube video (which didn't exist in the 90s when LANL wrote their lesson plan), here's a 4 part version of the tale sketched out in pen. Here's part one: https://www.youtube.com/watch?v=ZZ7ipq9T_bw and here's the video author's website: http://www.hotel-infinity.com/

Answer 9

One can give them some intuition.

Draw a line segment of length 'one'. Ask them how long it is.

Then draw a single point. Ask them how much room it takes up (the answer is zero).

Draw two points; how much room do they take up? (Zero) Keep drawing poiints one by one. Say, 'No matter how far I go, even if I go forever, to normal infinity, they take up no room.'

'So there must be an even bigger infinity of points in the line segment'.

Answer 10

To get playing with one-to-one correspondence, The Cat in Numberland, by Ivar Ekeland, is marvelous. Of course that only helps with ℵ0, but it's a step in the right direction. I wonder if there's a way to tell a story about ℵ1...

Answer 11

It's actually a very hard topic to teach. We don't want to teach them the Naive set theory definition of a cardinal number because Naive set theory is contradictory. In Zermelo-Fraenkel set theory, it's easy to define the property of there existing a bijection from one set to another set but the axiom of choice is not provable so it's very hard to define an object for each set such that two sets are assigned the same object when there's a bijection from one set to the other although that can be done using Scott's trick. Alternatively, you could create a provably consistent extension of ZF where you just declare that the cardinality of each set is a set such that two sets have the same cardinality when there is a bijection from one to the other. With either way of introducing them, you can show that there is in fact no set of all cardinal numbers.

Ordinal numbers are even easier to teach. First we have all the finite ordinals. Next comes $\omega$. Now starting from it, you can apply the successor function as many times as you want. The next larger ordinal is $\omega \times 2$. The supremum of all the ordinal numbers that can be expressed as $\omega \times$ a finite ordinal is $\omega \times \omega$. If you then again apply the successor function as many times as you want, the next ordinal after all of them is $(\omega \times \omega) + \omega$. We also have $(\omega \times \omega) + (\omega \times 2)$ We can keep going and get $(\omega \times \omega) + (\omega \times \omega)$ which can also be written as $(\omega \times \omega) \times 2$. We can also add to that any ordinal number less than $(\omega \times \omega) \times 2$. The supremum of all ordinal numbers that can be gotten that way is $(\omega \times \omega) \times 3$. The supremum of all ordinal numbers that can be expressed as $(\omega \times \omega) \times$ a natural number is $(\omega \times \omega) \times \omega$.

This is really a totally different concept of an ordinal number than the ZF definition and probably easier to teach than ZF. It includes only those ordinal numbers that are small enough to use to create a proof system of pure number theory. By Godel's incompleteness theorem, you can always think of proof systems corresponding to larger ordinal numbers than you could have before. Yet, those ordinal numbers will always be smaller than the Church-Kleene ordinal. How can that be when there's no end to constructing ordinal numbers useable to create a proof system? Because the Church-Kleene ordinal is unfathomably large. Once you create a system of ordinal numbers useable in a proof system, you still don't have all of the ones smaller than the Church-Kleene ordinal so it doesn't necessarily mean that when you make a major extension to that system, you will get one large enough to include the Church-Kleene ordinal. This last paragraph can probably only be understood by adults and probably is not worth teaching to elementry school kids. I wrote it just to explain how the concept of ordinal numbers they could be taught really is a completely different concept from the ZF definition and that concept is the one that should be taught.

Answer 12

Inspired by this answer and expanding on this one...

How many numbers are there? Infinity. (Write the first few naturals)

Can you tell me what an even number is? What are the first few? (Write them down) How many even numbers are there? Infinity!

Are there more numbers than there are even numbers? Yes, hmm, they're both infinity

But look, if I write the even ones under the ordinary numbers like this...

So we've proved that there are the same amount of even numbers as there are of all numbers. And we've also introduced the idea of infinite sets of numbers. So now we can say:

What other sequences of numbers can you think of that are infinitely long like this? All the numbers divisible by 3. All the numbers divisible by 4.

Good. What other kinds? All the square numbers. All the prime numbers. Just 0,1,0,1 repeating forever.

How many different infinite sequences of numbers do you think there are? Infinity.

The same infinity as the number of even numbers was? Sure, probably.

But look, if I write all the sequences you can think of alongside the ordinary numbers like this... (Writes Cantor's diagonal argument) Now I'm going to look down the diagonal and write a new sequence of integers by adding 1 to each... This sequence doesn't appear anywhere in the matching, does it? No.

And I can do this no matter what matching-up you try to do, right? Yeah, I guess.

So you'll never be able to match up all the sequences of integers with all the integers. No, there will always be at least one I didn't match up.

So (this is the leap) we say that the number of integer sequences is bigger than the number of integers. It's a bigger kind of infinity than the other kind.

Answer 13

My advice is to only do this for high track kids. And within this just teach topical awareness. I.e. just mention that there can be different sizes of infinities. And leave it at it. No not rigor. And no, not even intuition about it.

I think there is a sort of pedagogical blind spot that many math types here have, where the only way to learn something is in detail. Or worse, rigorous proof. But you can actually know about something, without knowing it in detail. Think of things like how a refrigerator works, that you have an idea on. But would get tripped up tracing temp entropy or describing the key components. It can be a nice little enrichment to know ABOUT something without knowing IT.

For example, I learned in a magnet school in 5th or 6th grade that there were different levels of infinity. And I never learned more than that. It was just an interesting assertion slash factoid of general knowledge. Like Bushmen of the Kalahari. To this date, that is my level of knowledge of the topic. Or consider the inability to solve general quintics. It was an aside in algebra class. Just magpie knowledge like the battle of Khe Sahn was in VN War. But no clue how or why USMC got tied down there. But at least I have the heard of it, level of awareness.