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I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. This could be useful to explain to a layperson the difference between countably and uncountably infinite sets.

Can anyone come up with a viable example?

Lonidard
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    Do you think "time" is continuous or discrete? What about space? – GEdgar May 23 '15 at 19:22
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    What do you mean by "reality"? – kjetil b halvorsen May 23 '15 at 19:23
  • You can count one dollar , two dollars .... etc You can even count one cent two cents ..... etc

    Imagine there is something smaller than a cent. call it $x$

    And another thing that is smaller than $x$ and so on, you will eventually reach the set of real numbers which is uncountable

     – alkabary May 23 '15 at 19:24
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    Even if the universe contained infinite many particels, the number would still be countable. The number of points on the interval $[0,1]$ is infinite uncountable, but the points cannot be marked in reality. – Peter May 23 '15 at 19:24
  • I believe that the infinite divisibility of time and space would lead to paradoxes. Zeno would most likely agree! – Lonidard May 23 '15 at 19:26
  • The concept of uncountability is quite difficult to understand. I still do not understand EXACTLY, why there are uncountable many real numbers. Cantor proves this with his famoud diagonalization, but I have difficulties to imagine a diagonal with infite many entries. – Peter May 23 '15 at 19:26
  • @bharb: What you believe is irrelevant if you want to explain a "fixed notion" (e.g. the definition of an uncountable set) to someone else. – Asaf Karagila May 23 '15 at 19:27
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    As far as I know, there is no object that is proven to be infinite, much less proven uncountably infinite. But then, there is no object for which we have really proven anything. – Thomas Andrews May 23 '15 at 19:28
  • @AsafKaragila True, indeed. My point is that I don't believe we can physically suppose that space and time are infinitely divisible while we can't prove it. – Lonidard May 23 '15 at 19:29
  • If you consider a quantum system with a continuous spectrum you could argue that the set of possible outcomes is uncountable and physically relevant. – hjhjhj57 May 23 '15 at 19:30
  • Zeno's paradoxes can be solved with finite sums with infinite many entries, even if time and space are continous. – Peter May 23 '15 at 19:30
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    We can't prove anything about the universe, so we shouldn't assume anything? We assume it because, in experiment ways, we find it is good enough to assume it. For example, calculus is really good at predicting the macro level behavior of the universe. So we use it, and assume these properties of space and time, not because they are true, but because they are close enough. But we cannot prove anything about space time, at all, just observe and conjecture and test. – Thomas Andrews May 23 '15 at 19:31
  • @ThomasAndrews That's a good point! – Lonidard May 23 '15 at 19:34
  • If you believe that any infinite decimal, like $13.4802356711043\ldots$, can be the length of a stick in inches then you have to accept that there is an uncountable number of reals. – Christian Blatter May 23 '15 at 19:36
  • That is pretty much clearly unprovable. @ChristianBlatter Belief isn't the question, and quite a lot of physicists are willing to admit this is possibly not the case. – Thomas Andrews May 23 '15 at 19:45
  • It's somewhat amusing that you presumably don't consider the reals to be realistic enough. – wchargin May 23 '15 at 22:29
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    @WChargin: They're not called "The really real reals", just "the reals" so there's no reason to expect that!!! :-D – Asaf Karagila May 23 '15 at 22:30
  • you could travel around the Earth countably infinite number of times, with certain unrealistic assumptions. That doesn't really answer your question but may be helpful to a layperson to understand you can pair off each circumnavigation with the natural numbers... – pshmath0 May 23 '15 at 23:21

5 Answers5

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That depends.

If by "realistic" you mean something that has to do with physical reality, then I defy you to come up with a set which has exactly $200^{200^{200}}$ elements.

If by "realistic" you mean something which comes up naturally in mathematics, then $\Bbb R$ is an uncountable set.

As for explaining the difference between them? That's not very easy, because first you need to be sure that the person understands the difference between sets of size $200^{200^{200}},200^{200^{200}}+1$ and $\aleph_0$. Which is most likely not going to be very easy. Sure, two of them have a finite number of elements, but it's so large it's infinite for all practical purposes. You couldn't even tell them apart if you put the two sets one right next to the other. If they can manage the difference there, then it's not difficult to explain what's "uncountable". Just infinite and not countable.

Unfortunately, mathematicians undergo a difficult training to work with definitions, rather than "common sense intuition" that we have before our studies. So explaining something that had to be earned by hard work is never easy. If it were, we wouldn't have to work so hard to get it.

Asaf Karagila
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If you accept that one can form infinite sequences from say the set of symbols $\{a,b\}$, then I would say that the set of all such sequences gives a fairly nice example of an uncountable set; as shown by Cantor's diagonal argument.

This may be simpler than the uncountability of the reals, or of the interval $[0,1]$ as you don't have to worry about some reals having more than one decimal representation.

paw88789
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We don't know if space is infinitely divisible, but if it is, then it has uncountably many points, because if $(x_n)$ is any infinite sequence of points, and if a sequence of regions of space $(S_n)$ is constructed recursively so that $S_{n+1} \subset S_n \setminus \{x_n\}$ ($n = 0, 1, 2, \ldots$), then this nested sequence contains at least one point that is not equal to $x_n$ for any $n$. (My use of set-theoretic notation is only suggestive. A. N. Whitehead and Jean Nicod attempted to define points of space as nested sequences of regions of space, taking regions rather than points as the fundamental concept.)

See the comments. This answer can't be taken as referring to actual physical space (according to our best scientific understanding); nor (intentionally) does it refer to any purely mathematical conception; therefore, at best it refers to a conceivable idea of physical space.

Calum Gilhooley
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  • Why can space be like $\Bbb Q$? – Asaf Karagila May 23 '15 at 22:29
  • @AsafKaragila Sorry, you've lost me! – Calum Gilhooley May 23 '15 at 22:32
  • Well, how do you define infinitely divisible? – Asaf Karagila May 23 '15 at 22:35
  • [Too late to edit previous comment:] The OP asked for an "object or realistic set" that can be shown uncountable. I take the notions of "region of space" and "point of space" as given (without prejudice as to whether points can be defined as equivalence classes of nested sequences of regions, as I recall Nicod tried to do), and take it as axiomatic that any nested sequence of regions contains at least one point. In a serious treatment (with or without 'point' as a primitive notion), one would need to state axioms to ensure that this is so. But the OP wasn't asking for a fully rigorous answer. – Calum Gilhooley May 23 '15 at 22:40
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    So, you're saying "Assuming that reality is a compact metric space without isolated points, it is uncountable". Sure, but that's a big assumption to make. And you might fare better writing this assumption explicitly. – Asaf Karagila May 23 '15 at 22:41
  • @AsafKaragila Just off the top of my head: "infinitely divisible" means that there exist non-stationary nested sequences of spatial regions (which might or might not be taken to define points of space, depending on how the topic is treated rigorously). Presumably the non-stationary requirement rules out such a sequence zeroing in on an "empty" part of space, because then it would be stationary. – Calum Gilhooley May 23 '15 at 22:44
  • @AsafKaragila Well, of course one could of course say anything of the form "something in reality is modelled by such-and-such a mathematical object, and here is a proof that said mathematical object is an uncountable set"! But I felt the OP was asking for an answer of a different character. Of course, I could be wrong! It's for the OP to say what he's really looking for. – Calum Gilhooley May 23 '15 at 22:46
  • Because I'd guess that infinitely divisible means just "dense" in the order theoretic sense. – Asaf Karagila May 23 '15 at 22:47
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    In theoretical physics the Planck length plays the role of a smallest length scale. – M. Wind May 24 '15 at 05:12
  • @M.Wind That makes my answer look as thick as two short Plancks. Physical facts are so inconvenient! – Calum Gilhooley May 24 '15 at 05:56
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    @M.Wind: The smallest distance that our current theories can deal with, you mean. Physics is only temporary until we find a better one. – Asaf Karagila May 24 '15 at 06:25
  • @AsafKaragila Indeed, I was trading not only on my own ignorance of physics, but also on collective uncertainty as to the ultimate nature of reality. But if our current best bet is that there is a lower limit to length, then it can hardly be said to be "proved", even in the loosest sense, that there are uncountably many points of space. I was trying to have my cake and eat it - treating "space" as physical, rather than purely mathematical (so that I didn't have to give a rigorous argument), but at the same time, ignoring inconvenient physical facts. So my answer won't do. – Calum Gilhooley May 24 '15 at 06:31
  • @AsafKaragila I was trying to say that in mathematics you can zoom in with arbitrary accuracy on the real numbers and find detail at each scale. See for example the magnificent Mandelbrot set. In physical space things may be different. There are reasons to assume that there might be a smallest scale, the Planck length. What happens at even smaller scales no-one really knows. – M. Wind May 24 '15 at 15:12
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I guess in a sense it depends if you believe the universe is inherently digital or analog. If analog, there is infinite detail, so the surface area of any actual physical object would be infinite.

Alan Wolfe
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Set of stars is infinite in reality. Infact it is countably infinite.

Another example of infinite set is "set of all points in a line segment is an infinite set"

See this it says the that set of integers is countably infinite whereas set of real numbers is uncountably infinite.

In countably infinite sets you cannot count the total number of elements but at least you can do the counting. Set of integers is an infinite set but you can count the elements in ascending or descending order(take any starting element).

In uncountably infinite sets you cannot even count because in between any two real numbers their are infinitely many reals. Here you cannot even do the counting! (suppose you are starting with 0 the you cannot find a number which is just greater then 0, if you think of 0.1 but 0.01 is closer to 0 to get more closer you have 0.001 and so on) this shows that you cannot do the counting.

(Remember you have to do the counting in either ascending or descending order without missing any element in between)

You can find a formal definition of uncountable sets here.

You can still compare the cardinality of two sets. If there exists a bijection between two sets then the cardinality of these two sets are equal. The cardinality of $\mathbb R$ and $\mathbb R^+$ is same! (See this)

Community
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Singh
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    Uhh, there are infinitely many stars? How do you figure? – Asaf Karagila May 23 '15 at 20:50
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    Yes, the infinitude of stars is a fairly large assertion to make. – Thomas Andrews May 23 '15 at 20:53
  • @AsafKaragila Sir see this video – Singh May 23 '15 at 20:59
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    Hrm, it says nothing about the infinitude of the stars there. In fact, all it says is that there are "probably" about septillion stars in the observable universe. That's a *very* finite number. – Asaf Karagila May 23 '15 at 21:05
  • @AsafKaragila Sir I have included another example in my answer please see that. I also want to say that number of stars are finite in "observable" universe but yet we haven't observed everything! – Singh May 23 '15 at 21:20
  • Yes, we didn't observe anything, but anything we cannot observe in any way is not provably there. So it would be the same as saying that "god is infinite". This is a matter of belief. The second example is again not something provable in physical reality and becomes a matter of belief on whether or not the universe is continuous or discrete. – Asaf Karagila May 23 '15 at 21:26