But before you do that it's worth just asking them what they think comes after everything else and see what they say. Because kids often have really interesting ideas on those kinds of topics and once you tell them something, then their ideas get pushed out.
This is one of my mottos for life - you can only be naive once so it can be beneficial to let your imagination run wild based on your unique background before you seek out best practices.
Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math!
The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewhere in that vicinity - and that's mathematics, the question then is - so what?
The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them.
The first time I've seen the notion of infinity was in a Russian children's book. There, they made a bijection between all the (infinite number) of points in a small segment and a larger segment - and even with all the points of an (infinite) line! I didn't really get it then; while I could find nothing wrong with the argument, it certainly looked like bullshit that a short segment could have as many points as a long one.
But some things don't have to make perfect sense right away.
The next time my understanding of infinity really improved (ignoring the notation for "x growing without bound" of calculus) was in the first year of college, with Cantor's diagonal argument. And I think that's when the picture from the book I read in kindergarten made sense, at last.
The bijection in that picture would have been boring if one could always make it. But with the diagonal argument, one sees that's not the case. That's what makes these infinities interesting and fun, to me.
So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. And really the kind you can explain to kids.
One night I've had a long tea on a rooftop of a Brooklyn apartment building with a friend who is an artist, and by the sunrise, she understood Cantor's diagonal argument - and enjoyed it.
It is quite regrettable that this is the kind of knowledge that's only generally shown to math majors in college. This is reason #712889 why we need to change the way we teach and talk about mathematics.
> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them.
There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful.
> So, I might be biased in that, but I think that the cardinals are the most playful type of infinity.
Definitely disagree. Once you know the basics, doing things with cardinals tends to be either boringly easy or impossibly hard. Ordinals, on the other hand, you can just play around with and actually get somewhere.
Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction!
>There's plenty you can do with ordinals too
Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
> Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction!
Sure, but do you really have to explain that?
When I imagine introducing kids to ordinals, like the OP talks about, I'm assuming it's basically taking the approach in, say, John Baez's blog posts on large countable ordinals[1].
> Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
Hm, maybe not. Maybe some of the classic examples of weird things that can happen with transfinite-time processes... but explaining any of that might be hard. And also that might not really be the right time to introduce people to discontinuity.
Really like I said I was basically thinking of the approach above, without application. I think it stands on its own pretty well, it's fun, you can play around with it -- it's basically the "is too, times infinity+1!" game except formalized (so it kind of comes naturally out of something kids already try to do) -- and the questions have actual answers.
I agree with you about Cantor's diagonalization. I think that really gives a "tactile" conception of infinity. If I had to explain infinity I'd probably choose to explain injections, surjections, and bijections; followed by countability and uncountability.
If the child is old enough to understand basic addition and multiplication, you can probably run through a short explanation of the different (elementary) number systems.
The cool thing is that you can introduce these concepts (bijections, etc) without calling them by their overly formal names, and yet maintaining rigor.
E.g.: on a pasture, there are black and white sheep. How can you know, without counting, whether there are more black sheep than white sheep, the other way around, or there's the same number?
Well, start taking them out in pairs, black and white sheep in each pair. If at some point you have a black sheep, but no white one to pair with, you know there at least as many black sheep as there are white ones. Same for the other way around. And if all sheep can come out in pairs like that, you conclude there must be the same number of them!
My wording is not the clearest here, but you can get the idea. The notion of "same size" for sets via putting things side-by-side is something kids can get before they learn numbers.
Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4 directions. Name the intersection points in order -- left to right, top to bottom.
What's the name of the first point in the first row? 0
What's the name of the first point in the second row? \omega
How is that more complicated or less interesting than cardinals?
Well, Cantor's diagonalization argument can be used to prove the Halting theorem, which is another problem that is easy to state and is very interesting to think about.
The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them.
But you can do a lot of math without ever having to deal with the ordinal numbers.
For that reason, cardinals are more interesting to me - not just as a concept in and of itself.
I guess I’m mixing up ordinals and cardinals but it seems odd that if you order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2), etc...)you never get to omega and cover all the same points.
Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.
EDIT: Just wanted to add that an order isomorphism has two requirements:
(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and
(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).
As a kid, I asked my mom if all numbers have a name, she said yes, and then I asked the obvious follow-up question: If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question.
That question was unresolved for me until recently, when I revisited the question and thought that at some point you could start naming them a, aa, aaa, aaaa, etc instead of seventen hexadecillions five hundred and sixty octillions ... or something like that.
I would love to remember all the other questions I had.
> If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question.
Ah, this is the philosophical distinction between every possible thing, and every conceivable thing.
The set of possible number names may be very large or infinite, but it is assembled from a small set of symbols like "one", "two" "seventeen" "million", "and", "times" and "three" "to the power of", etc.
A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols.
The set of the possible is a subset of the conceivable. Both sets might be infinite.
I think the distinction is really between "A given infinite set" and "every conceivable thing". It's a common fallacy to equate one infinity with any other infinity, and therefore any given infinity must contain all things. This fallacy is very neatly encapsulated by the phrase
>If there are infinite numbers, and all numbers have a name, then there must exist a number called door
> A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols.
Certainly you are able to represent "door" in a base-64 numbering system? (or base-36, or even base-28)
Ah, but "a base-64 numbering system" is a different set of rules ;)
And it's rules in which "door" is a "numbery name". And "Достоевский" is not.
You could still ask "If there are infinite base-64 numbers, and all base-64 numbers have a name, then must there exist a number called "Достоевский" ?" and the same argument against would apply.
The difference between "infinite" and "all possible" seems to be really unintuitive. The example I got to is that there are infinite even numbers, but not all natural numbers are even.
This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be odd universes.
I heard a really good interpretation of this recently.
In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter.
But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
Physically it’s possible for a ice cube to show up in the center of the sun. Each nucleus in the center of the sun is stripped it’s electrons but that’s the average view in shorter time frames they can have all of those electrons just very briefly. Three such atoms could collide just right and form a water molecule in the center of the sun. From their it’s just increasingly unlikely situations but that’s not enough in the face of unlimited random variations.
To play with that idea a little, imagine the ability to instantly teleport across space or even universes.
If you teleport a block of ice to coordinates which happen to be smack dab inside a star, then maybe for an instant, like a single "tick" of the processing engine, there would be an ice cube in the middle of the sun.
Why do you believe there is no sequence of events that could end with an ice cube in the middle of a sun?
One scenario I would not rule out is a large icy comet moving at large speed straight for a sun. Is there no combination of size and speed for which that would result in a small ice cube in the centre of a sun?
Hm, it would probably not be cube shaped though. But I agree, we could even calculate that one. It would certainly be an interesting spectacle.
Going back to the sequence of events argument, I think that implies that all universes agree on similar rules like ours (Physics), as we could only deduce this kind of argument with this assumption.
Maybe there is a universe where ice is scorching hot and all sun cores consist of it, and as matter cools it evaporates? Maybe there is even a universe where gravity and orbital mechanics work in a way to produce cube shaped stellar bodies?
Yes but there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun, and not just briefly, but for a substantial amount of time.
Yes this is the presumption. Remember infinity allows all possible skewing of our current physics and all possible permutations of it. Further reading: https://en.wikipedia.org/wiki/Multiverse
I internally-resolved this when I used to get [horribly, painfully] lost infinity back in in my acid-taking days with the short maxim "anything that can be, is".
If you're referring to real numbers, then virtually no real numbers have a name. If all real numbers had a name then you could order them alphabetically and put them in one-to-one correspondence with the natural numbers, which we know is impossible: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
to continue the discussion, there are many kinds of infinities, and some are larger than others.
one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....).
there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals.
as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity.
even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming.
* the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.
That is something I hit uppon recently, when I tried to find a binary representation that could concievably represent all real numbers, given enough bits of data. I don't think I've ever seen such a practical application of countability and cardinality. Really interesting where this stuff manifests in the real world.
Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.
[edit: for clarity i'm not going to use the term "name"]
the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings.
Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number.
for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place.
`+` :
technical details include that some numbers have non-unique representations as infinite sequences of digits.
For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going).
if this irritates you, let z = 0.999...
then 10 * z = 9.999...
then 10 * z - z = 9
then 9 * z = 9
so z = 1
I don't understand this explanation, and I'm an adult with an interest in mathematics and a degree in software development (although not mathematical per se, they usually go hand-in-hand). I like how "w" is the smallest number you cannot count to - if I understand it right it's almost 0 but not quite - but I don't understand the bonus questions and answers.
If this is the best "explain it like I'm 5" explanation of infinity, I believe I can think of a few examples that give a better idea of it. Heck, even the concept of "never ending" seems simpler to me. "Never ending" + 1 is still never ending.
I believe that approach comes naturally to us software developers, as we tend to think about mathematics in procedural terms (which is how programs run in the execution environment) rather than in string rewriting (which is how mathematical proofs are made). I also consider infinity as a process that never finds its ending condition, my mental model is the "infinite loop".
"ω" is not almost 0, that would be epsilon "ε". Omega ω is a number higher than any other natural number, i.e. "to the right" of the infinite line of numbers.
Mathematicians "will" this number into existence, so to say, starting from a contradiction. It's the same that they do with irrational numbers ("imagine there's this number "i" that, when multiplied by itself, it gives you -1"). With infinity, it's like: "you know this process that never ends? Well, imagine that it finishes, and let's call the result "ω".
Once they have this new number defined, they do lots of mathematical operations with it, trying to find its properties. What they never remember again after that is that the number dit not appear as the result of following the initial process to completion; they had to assume that it existed independently from the process.
* BTW, this is also why they have different kinds of infinities. They are using different never-ending processes in their respective definitions, and using the same name for all of them.
Notice that mathematicians don't take real numbers for granted too - we need to assume their existence first. Looking at this, complex numbers stop being that arbitrary, just another extension of reals.
Also, it's nitpicking, but i is imaginary, not irrational.
Also, complex numbers were accepted by mathematicians before negative numbers (it's something that boggles minds of some people).
A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth.
This is the opposite of what Feynman thought. He said, and I agree, something like "the hard part about teaching is making concepts simple without saying things that are false." The whole challenge of teaching is NOT lying, but instead saying simple things that are still true.
Feynman might have been sympathetic to the idea that all of science is a process of "replacing the lies with successively better approximations of the truth". Certainly Asimov was:
I like the concept, but if I may editorialize, I feel the phrasing needs work. "The smallest number you can't count to" is a negative statement, which makes it confusing right off the bat.
What do you mean, a number I can't count to? If I'm 8 years old this is like throwing a null pointer exception in my brain.
They might also take it literally. Like, they might think that infinity is less than a million because they know they couldn’t physically count to a million.
Haha I counted to 32,000 once when I was around 7/8 years old. All I remember is it taking multiple days and my mouth was dry and sore. Props to counting to a million!
If you count one a second, you're counting very slowly. Also, you can take breaks and pick it back up, making a note where you left off. I did it this way once, growing up.
> A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth.
Well put! I'm in the process of explaining my 3yo daughter different molecules (there are some cheap kits on Aliexpress) and cells and life on a micro scale and this description of moving from simple but inaccurate models to more complex and accurate is something that I've also noticed in my explanations.
Shudder. Consider a thought experiment - a military briefing. A captain briefing generals. One must necessarily simplify. But imagine a briefing that is grossly incomplete, assortedly incorrect, very misleading, written without understanding and without mentioning and characterizing that lack, and pervasively incompetently bogus, and that captain later on the carpet before the generals, defending the briefing with "well, all briefing is a process of lying". Shudder. I consider the "teaching is lying" meme to be vile. I've heard it most often associated with pre-college chemistry. Chemistry education research describes pre-college chemistry education content as "incoherent", leaving both students and teachers deeply steeped in misconceptions. Oh well.
You mentioned microscale, so I write just to note the top "How to remember sizes" section of my slowwwly-loading wasn't-intended-to-be-public page http://www.clarifyscience.info/part/Atoms . It might help you provide a framework for your kid to think about small things. FWIW.
> Shudder. I consider the "teaching is lying" meme to be vile.
Couldn't agree more. And when I read this in the article I also thought of high school chemistry even before I read your comment. I was put off chemistry in high school precisely because of its incoherence.
The best thing to do, as always, is to be honest. Tell your students that what you are teaching them is a simplification; a model that is useful at this level. Many will be satisfied with this answer and not want to go to a deeper level. But some children, those with a greater need for coherence and for things to make deep sense, will be reassured to know it's only a model and either wait for the deeper model to be taught or want to start exploring the deeper model at that time. Which is fine.
I found the approach taken by the author of this article quite patronising and rather distasteful.
My comment, and I believe the one I replied to, is directed at the first paragraph in this piece, which is about the idea of teaching as 'a process of lying' and one which the author agrees with 'in principle'.
I also do not like his suggested definition. It comes across as smartarse-ish and I think most children would feel the same way.
a military briefing is different from teaching. A military briefing should be communicating things the generals already understand.
teaching is a process of lying is kind of provocotaive, but teaching is a set of progressions, each progression necessarily leaves the edges blury while trying to make one aspect clear.
Pretty much. If generals are learning battlefield tactics and logistics from a captain, then the war is already lost.
There's a massive difference between sharing/communicating intelligence and teaching. The former relies on a common understanding of terminology, technique, etc, the later involves one side attempting to understand all of that to be able to facilitate the former.
I once read a critique of high school science textbooks which pointed out the great many "lies" that could be found even in the best of them - basically highly inaccurate or misleading descriptions and illustrations of basic scientific concepts. The thing is, though, that I recognized a great many of those same "lies" as being in my college science textbooks, too!
Yeah, this is pretty much exactly the kind of thing I am talking about. Except that the guy in question was generally replacing "relatively simple explanation which is more wrong than right" with "relatively simple explanation which is more right that wrong". Meanwhile the guy in the video seems to be adding quite a bit of complexity, even though he says that he's actually presenting a simplified view of the matter.
IIRC, one of the examples from the book thing had to do with the explanation and illustration of how light gets refracted when passing through glass. The overall effect is supposedly explained by the fact that light slows down as it enters glass, and typically a wavefront illustration is used to depict this. He went on to explain how the illustration is wrong and what the correct illustration should be instead, but I don't remember the details. And now that we have meta-materials which can bend light the wrong way, the whole "slowing down causes refraction" idea may be wrong, too. In fact, I remember reading an early potential explanation of this new effect (which as I recall had to do with the notion that the magnetic aspects of light might be affected differently than how the electrical aspects of light are, or vice versa) and thinking to myself "Aha - now that makes perfect sense!" And if that explanation makes sense for meta-materials then it also probably makes sense for regular materials. But once again I don't remember the details.
BTW, I understand the need to often simplify things quite a bit when you're dealing with students, but if we have "simple but incorrect" vs. "simple but correct" then we should be working hard to eliminate the former as soon as possible. If his critique of the situation is valid then it kind of beggars belief that we are still teaching so many things the wrong way!
Seriously? He gets this worked up explaining the difference between "the water is pulled out at the earth moon line" and "the water is pushed from the poles towards the earth moon line"? The net effect is identical.
If I understand his explanation correctly, the math says the effect can't be identical because the difference in gravitional force is much too small to lift the water that much. It has to be squeezed from the poles to raise the water high enough to account for tides.
> infinity is a useful made-up number that's bigger than all other numbers
I love the emphasis on this being a choice, and I wish this kind of thinking was taught more in math. So many math explanations act like these things are immutable facts of the universe, rather than human constructions. We lose some of the history and character of math when we teach it as law rather than invention.
Zero is also a useful made-up number, and it didn’t always exist as a concept. We understand 0 so deeply now, it’s not longer possible to imagine a world without that idea. Another fun one, 0^0 gets defined (chosen) to be 1 sometimes, because it makes other formulas work out, not because it’s the correct or only answer.
I wish I could upvote this more. I encounter this very often in discourse. I extend this to logic and philosophy as well.
Without maintaining both the rigor and the creativity of reasoning, only a hollow shell remains.
How I've come to view math is: all math is made up, parts of math seem to correspond with nature (physics), and can be used to build things (engineering).
A different bright kid might ask about ω−1, which opens a different but fruitful line of discussion: ω is not a successor ordinal, it is a limit ordinal.
ω - 1 makes sense in the surreal numbers (https://en.wikipedia.org/wiki/Surreal_number). The surreal numbers are the "greatest" ordered field, in a sense. They contain all ordinal numbers, which in turn contain all cardinal numbers. The cardinality of a set is just the least ordinal it can be put into a one-to-one correspondence with (this is called the von Neumann cardinal assignment).
I am not a mathematician, but I know that at least in standard mathematics it is fairly ill-advised to treat infinity implicitly as a number. Doing so can result in various contradictions (two different seemingly valid solutions to a problem). It should be thought of as a property of a process, i.e. for any number x you can change the process so that it results in a number greater than x. One example of infinity might be: "Think of a number which you are allowed to change after each number I suggest. If for any number I suggest you change it for a bigger one, then what you think of is infinity."
Chapter 15: Paradoxes of probability theory in Jaynes's "Probability Theory: Logic of Science" is a great reading on the topic (you can find a pdf easily on google). It starts with a quote from Gauss:
"I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit." -- C. F. Gauss
Anyway, there are plenty of theories in mathematics which use infinity implicitly, but one should perhaps be cautious.
Now you're conflating two different notions. You're (and your quotes) are talking about the first bullet point from the article. In turn, when taking about the last one, the one chosen by article author, from my experience, I'd say that it's beneficial not to even consider it in time, i.e. don't think about it as process, don't think of it as something completed. There's no completion, since there was no process to begin with. The object just is, and we can manipulate it according to the rules.
Many mathematically-trained people fumble the infinity question because infinity is just a term or notation for something that is non-ambiguously defined. People tend to view infinity as a sort of weird, maybe even mystic, concept. Certainly the intuition behind it is useful for reasoning, but mathematically-speaking it's just a term for another formal definition.
Consider limits. One can say that a function converges to a certain value v when its input approaches infinity. This seems all subjective and mystic and non mathematically trained people would come up with all sorts of interpretation of that. But this just means that no matter how small a number ε you chose, then there exists a number x such that for any input greater than x, the function's output will be contained within v-ε and v+ε. That's it. It's as simple as that. There is no mystery to it, nor judgment. "Infinity" is just a name that is involved in such formal property.
> Instead we can decisively say that there is another number after infinity, which is called “infinity plus one”.
I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else.
e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make it more confusing.
The article is a nice mind exercise but IMO not really helpful in explaining infinity to a child.
You are correct. Counting here changes its meaning.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
What I hear you say is that you can count to finite numbers in finite time, and infinite numbers in infinite time. That doesn't help me in understanding infinity.
The key point here is that you shouldn't be thinking of counting as a process that happens in time. How is infinity (here, omega) defined? We just say that it comes after any other natural number. So how to think about it? You shouldn't tell yourself:
- ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ?
You should be telling yourself
- ok, take number n. It's smaller than omega. Now check any other number m. If this property holds for any given number, then omega is greater than any given number, and thus infinite.
It's two different processes. Normal counting is saying, "take a thing and add one" (a successor ordinal). It's how we get from 42 to 43. If a number can be reached from zero using just this kind of counting, we call it finite.
The new type of counting also allows you to take a different step and say, "take a collection of things and add a new thing at the end" (a limit ordinal). It's how you get from the set of finite numbers to ω. There's nothing that's "one before" ω, but all the finite numbers are before it.
Repeat these steps as much as you want (making sure none of your sets are circularly defined), and you're counting with ordinals.
I think Conway and Guy's "The Book of Numbers" might be a better source. It's more focused on infinities as different types of number, and certainly at a more suitable level. ONAG is quite a technical book; children don't really need to know that the surreals are the universally embedding real-closed field.
Another approach is taken in the comic "Life on the Infinite Farm" (pdf: https://www.math.brown.edu/~res/farm.pdf). I'm not sure it's entirely appropriate for children though.
I got excited because I thought that I might finally understand infinity. Unfortunately I am still not there as I am not there with quantum mechanics. I have read many books on both with equal non-success.
A very nice book in French is "Le Chat au pays des Nombres" by Ivar Ekeland. The book tell a story that happens in the infinite hotel on the numbers planet. This hotel managed by the Hilbert is always full. It is a very good approach of infinity for children around 6.
When my daughter was learning fractions, it was fun to use the infinity to discuss the real numbers between 0 to 1 by squeezing 1/<whatever number she named> into them.
I like to think about the length of the outside of a circle. It's only when you use a line with a beginning and end in the shape of a circle that you can measure it. The idea can be extended to 3D with a spiral staircase that has no beginning and end. The steps of the staircase cannot be measured until you give it limits.
This is interesting to read as I've been struggling with the best approach for my son who's coming up for five.
He has a pretty solid understanding of basic addition, subtraction and multiplication.
He was asking about infinity and although he knows it's very big he was still asking about infinity plus one. Maybe it's not too soon to explain the concept.
What came to my mind was "a number is where to stop counting up. e.g. I'll walk for five blocks. I'll eat ten jelly-beans. I'll throw the ball twenty times. Infinity mean never ever stopping."
but that's much the same as "It's the smallest number you can't count to."
most of these comments are picking apart the explanation, but honestly this is a great way to introduce kids to the concept and have them "arrive" at the concept of infinity through their own logic.
> A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth.
In philosophy of sciences, it is called "idealization" and "concretization". Many philosophers from Poznan even name this philosophy of science as "Marxist philosophy of science", since Marx uses the technique of idealization and concretization in his work "Das Kapital".
Just ask them to do i++ and never stop. That is also infinity.
(or in simpler words, imagine you have a table than you put one apple to it, then one again, amd again and again and never stop. And yes, they would also soon understand, that the table needs to be infinitly big)
edit: of course it is about the sum of the process. they should imagine the pile of things or the number if you never stop adding.
That's an infinite process (which is an awkward way to describe anything), but not infinity.
You'll get more and more natural numbers, but will never actually get infinity.
I love ops explanation because it gives more natural picture: there are _other_ numbers beyond naturals. Then you tell kids there are also negative numbers (integers) and rational numbers and real and complex and so on.
I believe that way it is much easier to grasp that very abstract (but fundamental) idea of 'number'. That it is not just 1,2,3...
Because otherwise, we get people thinking "Complex numbers do not exist, that is just a silly thing used by mathematicians that has nothing to do with the real world, it is useless."
I think the problem with people thinking complex numbers don't exist actually stems from the imaginary component and the same problem of imaginary numbers, which is rooted not in failing to teach about other kinds of numbers but in the name. Especially given the name of the real numbers.
The idea that real numbers are real, imaginary numbers are not real, and complex numbers that have a real number part and an imaginary number part are also not real is a natural consequence of unfortunate naming choices.
For me, back in school days (or was it university?) it was a revelation when I came across quaternions. It suddenly clicked. I finally understood that there was nothing special about complex numbers (despite their special names and weird look). It was just an extension and a very intuitive one!
I finally understood that real numbers are the same thing:
tuples. They just happen to have exactly one element, hence we omit parens and everything else and just write that element (number)!
Complex numbers have 2 elements (real and imaginary). Quaternions - four. And so on.
> I finally understood that real numbers are the same thing: tuples. They just happen to have exactly one element, hence we omit parens and everything else and just write that element (number)! Complex numbers have 2 elements (real and imaginary). Quaternions - four. And so on.
Yes, but if you go too far with thinking this way it becomes easy to confuse complexes, quaternions, octonions, etc with 2-,4-, and 8-dimensional vectors. The additive and multiplicative behavior of complexes, etc changes in increasingly pathological ways as you increment the dimension. This is not the case with vectors.
In many cases you can safely replace R^2 with C. But there are specific cases where you cannot, because treating the complexes as just a pair of real numbers doesn't work the same way as if it was just a vector. Differentiable functions come to mind because they behave differently in R^2 than C, and when you're working with rings (instead of fields and vector spaces) they are also different. In a lot of places the isomorphism between R^2 and C is actually a happy accident rather than a definition.
And I thought we are talking about kids? There are not many kids who would understand the professors way.
So isn't it first about making them understand the concept of infinity? That they can imagine it, before you bombard them with other abstract math concepts?
I think that describing infinity as an infinite process is bad because that way we are basically saying that this infinite process and infinity is the same thing!
I love ops explanation because it separates infinity (as a number) and infinite process (counting).
> Just ask them to do i++ and never stop. That is also infinity.
Thinking back to that bug last week, where someone forgot that you can't just compare ever-incremented sequence numbers with 'x < y'... Maybe we don't want to encourage that idea :)
Square One TV had a brilliant take on this. One of their cast (of the main show, not Mathnet) would say something like: "I've thought of the biggest number: 973,281,472!" At which point the rest of the cast would respond (exasperatedly, in unison): "Add one more to it!" Really drove home the point that no matter how big a number you thought of, you could always produce a bigger number by adding one to it.
>Just ask them to do i++ and never stop. That is also infinity.
What, specifically, is infinity here? You are being vague here. Is it the process of counting up and never stopping? (that's the calculus infinity) The number you get in the end? (that's the infinite ordinal, but that is, as you noted, complicated) The number of numbers? (that's the cardinality of the set of integers)
Note that you can't say that the infinity is all that. The infinities I defined here are all radically different notions!
>In simpler words, imagine you have a table than you put one apple to it, then one again .. and never stop. [..] they would also soon understand that the table needs to be infinitely big
That's another infinity here - we're talking about size of geometric object here.
For starters, what happens if each next apple is half the size of the previous one? (Your assumption is false then!).
With equally-sized apples, it's still close enough to the notion of cardinality.
The problem with this explanation is that it doesn't help understand the nature of infinity, so it's not an explanation really - but a good start.
Some questions to explore from here:
1. Do that process twice. Now you have two infinitely big tables, but each has an end. Put them together end to end. Did you just get a longer table? Are there more tables on this combined table than you had on just one table?
2. Perhaps after you're done putting apples on the table, you realize that one apple rolled under the table. The table is absolutely full, but you want to put another apple on it. What do you do?
3. You find a pile of apples, each having a label consisting of a rational number on them. You notice that no two apples have the same label. Will they fit on the table? Can you put them without removing any apples already them? (The answer here is, mind-bogglingly, yes!)
4. Same question, but now the labels are infinite strings. (The answer is no, by the way, but it's by no means ovbious![3])
Etc, etc, etc... Essentially, Hilbert's Grand Hotel[1][2] with apples. I highly recommend reading this to get a decent understanding of one of the infinities.
I find explanations like this dangerously misleading, like Hilbert's Hotel and the Banach-Tarski paradox discussion talking about peas and suns. Infinity is not finite. Making it pretend-finite by making impossible analogies to finite physical objects instills fundamental misunderstanding of what finite and infinite means.
Answer to your question number 3 is: well, if I have just one apple, with label 3.1415926535 then obviously it will fit. Notice that no two apples have the same labels.
