13 ms·

How to explain infinity to kids

182 points by gbacon 8y ago

Boulth 8y ago
> 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.
- southern_cross 8y ago
  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!
  - rrauenza 8y ago
    Like tides? What Physics Teachers Get Wrong About Tides! | Space Time | PBS Digital Studios: https://www.youtube.com/watch?v=pwChk4S99i4 https://www.youtube.com/watch?v=pwChk4S99i4
    - pilom 8y ago
      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.
      - geomark 8y ago
        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.
    - southern_cross 8y ago
      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!
- analog31 8y ago
  All models are wrong, but some models are useful -- George Box
- mncharity 8y ago
  >> all teaching was a process of lying > Well put! 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 http://www.clarifyscience.info/part/Atoms . It might help you provide a framework for your kid to think about small things. FWIW.
  - oska 8y ago
    > 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.
    - vilhelm_s 8y ago
      The approach in this article is to give a "technically unimpeachable" definition. It doesn't involve any lies.
      - 8y ago
mbrumlow 8y ago
THis is one of the best things I have read on HN lately! I can't wait to use his answer to what infinity is to my kid when he gets old enough to ask such a question.
deleted 8y ago
[deleted]
raincom 8y ago
> 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".
gyrfalcon 8y ago
Infinity is a mathematical concept that doesn't actually exist. Just like the infinity gauntlet, it's a made up imaginary thing. https://plus.maths.org/content/does-infinity-exist https://plus.maths.org/content/does-infinity-exist
hutzlibu 8y ago
Why so complicated? 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.
- bitwize 8y ago
  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.
- kovrik 8y ago
  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."
  - dragonwriter 8y ago
    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.
Sniffnoy 8y ago
Nitpicking: This article writes 2⋅ω where what's meant is ω⋅2.
romwell 8y ago
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.
- antidesitter 8y ago
  I agree. Cardinal numbers (and bijections) are probably easier to start with.
  - zhte415 8y ago
    Given Cantor made it for the battle field.. it needed to be easy to deal with.
skate22 8y ago
I like the way I was taught better: Imagine you are 2 feet from the wall, and with every step you move forward 50% of the remaining distance. How many steps does it take to the wall. And the answer of course is that you never get to the wall, no matter how many steps you take
- throwawaymath 8y ago
  That ceases to be a paradox once you distinguish between boundedness versus completeness for a set of infinite steps. There are infinitely many steps on the interval [0,1]. But if you add 1 "step" at any point of the interval greater than 0, you've still "passed" it.
antidesitter 8y ago
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 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).
User23 8y ago
John Conway's On Numbers and Games has a great overview of infinities. And since it's about games it'll be great for teaching kids too!
- OscarCunningham 8y ago
  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.
- yiyus 8y ago
  Another book I really liked that talks about infinity is "A Certain Ambiguity", by Gaurav Suri and Hartosh Singh Bal.
scandox 8y ago
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.
- jackgolding 8y ago
  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.
peteretep 8y ago
If we're going for useful lies to tide kids over, I quite like the explanation in the Postgres docs: > infinity (date, timestamp) later than all other time stamps I think you could tell a kid who wasn't quite ready for Aleph numbers that: > infinity is a useful made-up number that's bigger than all other numbers Which is useful when kids first hear about it, I guess. I think my first "practical" introduction to infinity came with "Space is infinitely big", and both mjd's explanation and my own fails at that point.
- dahart 8y ago
  > 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.
  - KennyCason 8y ago
    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.
  - navane 8y ago
    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).
Izmaki 8y ago
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.
- oska 8y ago
  > Heck, even the concept of "never ending" seems simpler to me. This is actually the definition that the Simple English wikipedia article [1] gives. The very first sentence in that article: > Infinity is about things which never end. [1] https://simple.wikipedia.org/wiki/Infinity https://simple.wikipedia.org/wiki/Infinity
- TuringTest 8y ago
  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.
duncancarroll 8y ago
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.
- kylec 8y ago
  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.
  - marcelluspye 8y ago
    >couldn’t physically count to a million. This is only because they/you haven't tried. Source: counted to a million once.
    - lwansbrough 8y ago
      How long did that take you? Seems like it take at least a week.
      - kirubakaran 8y ago
        A million seconds is over 11 days straight without stopping for sleep, food, or anything! I very much doubt that they actually did it.
        cyborgx7 8y ago
        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.
        udkyo 8y ago
mirimir 8y ago
> It's the smallest number you can't count to. I don't remember whether I was ever taught that. But I do remember the line from Marilyn Manson's "Posthuman". > God is a number you cannot count to.
JeanMarcS 8y ago
Teaching mathematics is lying to kids all along, at least in France : - No, you can’t do 2-3. If you have 2 apples, you can’t give 3 - Well, in fact you can, it’s negative numbers. But if you have 5 apples you can’t split them equally between 2 persons. - Well, in fact you can. They got 2.5 each. But you cannot solo 2 apples in 3 equally !! - Well in fact you can. That’s 2/3 each. - etc...
- likeclockwork 8y ago
  > - No, you can’t do 2-3. If you have 2 apples, you can’t give 3 Well, how soon is too soon to teach children about debt?
  - JeanMarcS 8y ago
    Apparently the 18th century :-D ! https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid...
bronlund 8y ago
I'm glad you guys didn't try to explain this to me when I was a kid. Infinity is that which never ends.
nicodjimenez 8y ago
\infty = \lim _ { 0 \to \infty } x that's how
secabeen 8y ago
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.
- gowld 8y ago
  You don't even need the reals for that. Rationals is plenty.
Aeolun 8y ago
I don’t know if this would make any sense to kids, it certainly doesn’t to me. If infinity + 1 exists, then that number would be infinity. It’d be more accurate to say that Infinity + 1 = NaN
peter30744 8y ago
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.
- deleted 8y ago
  [deleted]
- enedil 8y ago
  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.
swfsql 8y ago
(related - and I enjoyed the vibe he put into it): https://www.youtube.com/watch?v=SrU9YDoXE88 https://www.youtube.com/watch?v=SrU9YDoXE88 - Vsauce's "How to Count Past Infinity"
tzs 8y ago
Or you could raise them as ultrafinitists [1]! [1] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism
avip 8y ago
Just give them two mirrors and then play.
mhneu 8y ago
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. That said, I see the appeal of choosing omega.
- n4r9 8y ago
  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: https://chem.tufts.edu/answersinscience/relativityofwrong.htm https://chem.tufts.edu/answersinscience/relativityofwrong.ht...
mchahn 8y ago
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.
deleted 8y ago
[deleted]
sattoshi 8y ago
I loved how a math teacher I knew talked about infinity: it is a garden where they grow very large numbers. It communicates that you can't work with infinity like with a number, because it is a group of numbers. It swallows all the operands as infinity does: 2 groups = still a group; half a group is still a group; a group plus one is still a group, etc. Then you have groups of groups, or infinity powers which work the same way. A group of groups is clearly greater than just a group. At least, I think this is what he meant. He never really expanded on it beyond the garden analogy.
DomreiRoam 8y ago
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. [1] https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-Nombres/434839 https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-N...
nurettin 8y ago
So we use a symbol and when they ask what it is, we use recursion to demonstrate the idea.
abhchand 8y ago
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.
sandov 8y ago
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.
- jpulgarin 8y ago
  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 https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
  - cyborgx7 8y ago
    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.
  - wwweston 8y ago
    Does this apply if names are infinitely long?
    - cyborgx7 8y ago
      No, then the names could just be the actual decimal representation of the number. But strings are in genereal considered to be finite.
    - 8y ago
YouAreGreat 8y ago
> It's the smallest number you can't count to Funny. If explaining infinity to a kid had been my problem, I'd have tried hard to avoid calling infinity a number.
SublimeWarior 8y ago
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.
tomclive 8y ago
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.
dnate 8y ago
> 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.
- navane 8y ago
  “Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
  - romwell 8y ago
    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. And now you're counting in a new way. [1]https://www.mathsisfun.com/activity/count-billion.html https://www.mathsisfun.com/activity/count-billion.html
    - navane 8y ago
      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.
      - enedil 8y ago
        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.
SideburnsOfDoom 8y ago
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."
lngnmn 8y ago
It does not exits, it is only a mathematical abstraction, based on abstract notion of a Number, which also does not exist outside your head.
axilmar 8y ago
What an awful way to explain infinity to kids and adults. How about something simpler? like 'infinity is when something never ends'?
- enedil 8y ago
  Your explanation is worse, since it doesn't allow to distinguish different infinities.
  - axilmar 8y ago
    It doesn't have to explain different infinities. When we talk about infinity, we need to realize the concept of 'never ending', and not anything else.
magerleagues 8y ago
Wait, why can't you count to it?
injb 8y ago
From the article: “Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” How is 'adding one more' not the same thing as counting?
Pete_D 8y ago
Another approach is taken in the comic "Life on the Infinite Farm" (pdf: https://www.math.brown.edu/~res/farm.pdf https://www.math.brown.edu/~res/farm.pdf). I'm not sure it's entirely appropriate for children though.
enturn 8y ago
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.
asdffdsa321 8y ago
I thought infinity was most always described in terms of limits. I have trouble thinking about it as a number

182 points by gbacon 8y ago

> 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.

southern_cross 8y ago
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!
- rrauenza 8y ago
  Like tides? What Physics Teachers Get Wrong About Tides! | Space Time | PBS Digital Studios: https://www.youtube.com/watch?v=pwChk4S99i4 https://www.youtube.com/watch?v=pwChk4S99i4
  - pilom 8y ago
    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.
    - geomark 8y ago
      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.
  - southern_cross 8y ago
    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!
analog31 8y ago
All models are wrong, but some models are useful -- George Box
mncharity 8y ago
>> all teaching was a process of lying > Well put! 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 http://www.clarifyscience.info/part/Atoms . It might help you provide a framework for your kid to think about small things. FWIW.
- oska 8y ago
  > 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.
  - vilhelm_s 8y ago
    The approach in this article is to give a "technically unimpeachable" definition. It doesn't involve any lies.
    - 8y ago

mbrumlow 8y ago

THis is one of the best things I have read on HN lately! I can't wait to use his answer to what infinity is to my kid when he gets old enough to ask such a question.

deleted 8y ago

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raincom 8y ago

> 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".

gyrfalcon 8y ago

Infinity is a mathematical concept that doesn't actually exist. Just like the infinity gauntlet, it's a made up imaginary thing. https://plus.maths.org/content/does-infinity-exist https://plus.maths.org/content/does-infinity-exist

hutzlibu 8y ago

Why so complicated? 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.

bitwize 8y ago
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.
kovrik 8y ago
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."
- dragonwriter 8y ago
  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.

Sniffnoy 8y ago

Nitpicking: This article writes 2⋅ω where what's meant is ω⋅2.

romwell 8y ago

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.

antidesitter 8y ago
I agree. Cardinal numbers (and bijections) are probably easier to start with.
- zhte415 8y ago
  Given Cantor made it for the battle field.. it needed to be easy to deal with.

skate22 8y ago

I like the way I was taught better: Imagine you are 2 feet from the wall, and with every step you move forward 50% of the remaining distance. How many steps does it take to the wall. And the answer of course is that you never get to the wall, no matter how many steps you take

throwawaymath 8y ago
That ceases to be a paradox once you distinguish between boundedness versus completeness for a set of infinite steps. There are infinitely many steps on the interval [0,1]. But if you add 1 "step" at any point of the interval greater than 0, you've still "passed" it.

antidesitter 8y ago

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 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).

User23 8y ago

John Conway's On Numbers and Games has a great overview of infinities. And since it's about games it'll be great for teaching kids too!

OscarCunningham 8y ago
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.
yiyus 8y ago
Another book I really liked that talks about infinity is "A Certain Ambiguity", by Gaurav Suri and Hartosh Singh Bal.

scandox 8y ago

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.

jackgolding 8y ago
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.

peteretep 8y ago

If we're going for useful lies to tide kids over, I quite like the explanation in the Postgres docs: > infinity (date, timestamp) later than all other time stamps I think you could tell a kid who wasn't quite ready for Aleph numbers that: > infinity is a useful made-up number that's bigger than all other numbers Which is useful when kids first hear about it, I guess. I think my first "practical" introduction to infinity came with "Space is infinitely big", and both mjd's explanation and my own fails at that point.

dahart 8y ago
> 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.
- KennyCason 8y ago
  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.
- navane 8y ago
  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).

Izmaki 8y ago

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.

oska 8y ago
> Heck, even the concept of "never ending" seems simpler to me. This is actually the definition that the Simple English wikipedia article [1] gives. The very first sentence in that article: > Infinity is about things which never end. [1] https://simple.wikipedia.org/wiki/Infinity https://simple.wikipedia.org/wiki/Infinity
TuringTest 8y ago
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.

duncancarroll 8y ago

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.

kylec 8y ago
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.
- marcelluspye 8y ago
  >couldn’t physically count to a million. This is only because they/you haven't tried. Source: counted to a million once.
  - lwansbrough 8y ago
    How long did that take you? Seems like it take at least a week.
    - kirubakaran 8y ago
      A million seconds is over 11 days straight without stopping for sleep, food, or anything! I very much doubt that they actually did it.
      - cyborgx7 8y ago
        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.
        udkyo 8y ago

mirimir 8y ago

> It's the smallest number you can't count to. I don't remember whether I was ever taught that. But I do remember the line from Marilyn Manson's "Posthuman". > God is a number you cannot count to.

JeanMarcS 8y ago

Teaching mathematics is lying to kids all along, at least in France : - No, you can’t do 2-3. If you have 2 apples, you can’t give 3 - Well, in fact you can, it’s negative numbers. But if you have 5 apples you can’t split them equally between 2 persons. - Well, in fact you can. They got 2.5 each. But you cannot solo 2 apples in 3 equally !! - Well in fact you can. That’s 2/3 each. - etc...

likeclockwork 8y ago
> - No, you can’t do 2-3. If you have 2 apples, you can’t give 3 Well, how soon is too soon to teach children about debt?
- JeanMarcS 8y ago
  Apparently the 18th century :-D ! https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid...

bronlund 8y ago

I'm glad you guys didn't try to explain this to me when I was a kid. Infinity is that which never ends.

nicodjimenez 8y ago

\infty = \lim _ { 0 \to \infty } x that's how

secabeen 8y ago

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.

gowld 8y ago
You don't even need the reals for that. Rationals is plenty.

Aeolun 8y ago

I don’t know if this would make any sense to kids, it certainly doesn’t to me. If infinity + 1 exists, then that number would be infinity. It’d be more accurate to say that Infinity + 1 = NaN

peter30744 8y ago

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.

deleted 8y ago
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enedil 8y ago
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.

swfsql 8y ago

(related - and I enjoyed the vibe he put into it): https://www.youtube.com/watch?v=SrU9YDoXE88 https://www.youtube.com/watch?v=SrU9YDoXE88 - Vsauce's "How to Count Past Infinity"

tzs 8y ago

Or you could raise them as ultrafinitists [1]! [1] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism

avip 8y ago

Just give them two mirrors and then play.

mhneu 8y ago

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. That said, I see the appeal of choosing omega.

n4r9 8y ago
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: https://chem.tufts.edu/answersinscience/relativityofwrong.htm https://chem.tufts.edu/answersinscience/relativityofwrong.ht...

mchahn 8y ago

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.

deleted 8y ago

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sattoshi 8y ago

I loved how a math teacher I knew talked about infinity: it is a garden where they grow very large numbers. It communicates that you can't work with infinity like with a number, because it is a group of numbers. It swallows all the operands as infinity does: 2 groups = still a group; half a group is still a group; a group plus one is still a group, etc. Then you have groups of groups, or infinity powers which work the same way. A group of groups is clearly greater than just a group. At least, I think this is what he meant. He never really expanded on it beyond the garden analogy.

DomreiRoam 8y ago

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. [1] https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-Nombres/434839 https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-N...

nurettin 8y ago

So we use a symbol and when they ask what it is, we use recursion to demonstrate the idea.

abhchand 8y ago

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.

sandov 8y ago

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.

jpulgarin 8y ago
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 https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- cyborgx7 8y ago
  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.
- wwweston 8y ago
  Does this apply if names are infinitely long?
  - cyborgx7 8y ago
    No, then the names could just be the actual decimal representation of the number. But strings are in genereal considered to be finite.
  - 8y ago

YouAreGreat 8y ago

> It's the smallest number you can't count to Funny. If explaining infinity to a kid had been my problem, I'd have tried hard to avoid calling infinity a number.

SublimeWarior 8y ago

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.

tomclive 8y ago

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.

dnate 8y ago

> 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.

navane 8y ago
“Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
- romwell 8y ago
  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. And now you're counting in a new way. [1]https://www.mathsisfun.com/activity/count-billion.html https://www.mathsisfun.com/activity/count-billion.html
  - navane 8y ago
    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.
    - enedil 8y ago
      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.

SideburnsOfDoom 8y ago

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."

lngnmn 8y ago

It does not exits, it is only a mathematical abstraction, based on abstract notion of a Number, which also does not exist outside your head.

axilmar 8y ago

What an awful way to explain infinity to kids and adults. How about something simpler? like 'infinity is when something never ends'?

enedil 8y ago
Your explanation is worse, since it doesn't allow to distinguish different infinities.
- axilmar 8y ago
  It doesn't have to explain different infinities. When we talk about infinity, we need to realize the concept of 'never ending', and not anything else.

magerleagues 8y ago

Wait, why can't you count to it?

injb 8y ago

From the article: “Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” How is 'adding one more' not the same thing as counting?

Pete_D 8y ago

Another approach is taken in the comic "Life on the Infinite Farm" (pdf: https://www.math.brown.edu/~res/farm.pdf https://www.math.brown.edu/~res/farm.pdf). I'm not sure it's entirely appropriate for children though.

enturn 8y ago

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.

asdffdsa321 8y ago

I thought infinity was most always described in terms of limits. I have trouble thinking about it as a number