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I am taking a course in mathematics that covers countability. The trick with the uncountability of the real line is that no matter how many times you divide up an interval, there would still be a real number inside of that interval so that even the smallest interval contains "more than infinite" amount of numbers.

However it seems that our "reality" as we experience it on Earth behaves nothing like the real line. For example, the total number of blades of grass in a soccer field is countable. Similarly, the cardinality of all the sand on the beach is also countable. Ants, leaves, anything macroscopic. Not just countable but finite.

But what about things that are smaller. For example, numbers of thermally excited electrons over a surface, or number of photons hitting our retina? What are some things that can be truly considered uncountable within our universe?

(Obviously excluding the multiverse hypothesis)

Emilio Pisanty
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Fraïssé
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    As soon as somebody builds a machine that counts to infinity and back we will be smarter. Until then I would consider everything either countable or irrelevant. – CuriousOne Jun 12 '16 at 02:54
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    This is a wonderful question, but I think unanswerable. To know something is uncountably infinite is to have a falsifiable test of its continuity. The bedrock of science is observation, and I don't know how you can demonstrate something is definitely continuous, not discrete but just beyond precision measurement. – anon01 Jun 12 '16 at 02:57
  • As other people have pointed out, your argument that the reals are uncountable does not work: I think you need to use something like Cantor's diagonal trick instead. –  Jun 12 '16 at 10:27
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    "The trick with the uncountability of the real line is that no matter how many times you divide up an interval, there would still be a real number inside of that interval so that even the smallest interval contains "more than infinite" amount of numbers." That's still true of the rationals, which are countable. – David Richerby Jun 12 '16 at 11:54
  • @CuriousOne. I see you are a man of physics. – Star Alpha Jul 28 '21 at 03:55

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The fact that you always find a number in between two other numbers is not exclusive of uncountable sets, like the reals. Rational numbers are countable and have that property too. All the examples you used are countable. Both real numbers and rational numbers are not well ordered sets on their standard order (the ordering in the real line).

In physics we use uncountable sets of at most $\aleph_1$, the cardinality of the real numbers. They appear naturally in our description of space. There are some people that propose that space is best described by either a countable set, or a space of higher cardinality, such as the surreal numbers. But for now the real line works fine. Most variables in physics that are defined or related to space (likely perhaps most quantities in physics, such as forces, energy, temperature, etc) are of uncountable cardinality, because the uncountability of the real line permeates through them. But many other variables are countable, such as number of particles, etc. However I do not find myself capable enough to give you a thorough list.

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    We don't use any uncountable sets in physics. We pretend to use math that does define such sets, but we really do not care at any time about the details. All of the physics ever possible can be done without any loss of accuracy with integers of probably no more than roughly 100 decimals in length, of which all but the leading 20 will always be in the noise. One can probably argue that the 80 bit IEEE floating point format was added because in physics the 64 bit format is too short, but the 80 bit format is mostly good enough. – CuriousOne Jun 12 '16 at 03:53
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    @CuriousOne I agree with you in that all physics can be described by a countable number of reals, if that is what you mean. Just notice that it is easier to treat space as the continuum. Many sums become integrals, etc. –  Jun 12 '16 at 04:31
  • I completely agree that the real numbers are a godsend (well... sent by a century of mathematicians, anyway), but they kind of detract a little from what physics really does, which is to compare physical quantities to other, more easily standardized physical quantities. The latter process, if taken literally, would be horribly pedantic, of course, so we borrow stuff from the mathematicians to fake it real good. :-) – CuriousOne Jun 12 '16 at 04:35
  • $\text{ZFC} \nvdash 2^{\aleph_0}=\aleph_1 \land \text{ZFC} \nvdash 2^{\aleph_0}>\aleph_1$ (in other words, the continuum hypothesis is independent of ZFC). Therefore it is bad practice of assuming that as true without specifying the logic model you're in. If you use reals, you are using sets up to cardinality $2^{\aleph_0}$, in general not $\aleph_1$. – yuggib Jun 13 '16 at 06:31
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    @yuggib Of course, but I do not think that level of detail adds to this kind of answer –  Jun 13 '16 at 06:36
  • and I don't see why you work in $\text{ZFC}+\text{CH}$ while everybody in physics simply works with $\text{ZFC}$; it is a non-standard choice that just complicates things... ;-P – yuggib Jun 13 '16 at 06:37
  • I believe in CH, at the religious level, haha, –  Jun 13 '16 at 06:39
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Your argument for the uncountability of the real line does not work, as it would show that the rationals are uncountable.

Both the number of blades of grass and of grains of sand are finite, so they have nothing to do with infinity.

You say that reality "behaves nothing like the real line"; yet, calculus is used to put a rocket on the moon, or to do a flyby of Pluto, and a million more very real applications.

Do I claim that the reality we perceive is uncountable? Not at all. In fact, it would be impossible to test such hypothesis, because all our measurements are very finite. But the whole of physics, amazing as it is, is nothing but a mathematical model. In the reality we measure there are no points, no axis, no vectors, no Hamiltoneans, no wave equations, etc., etc. These are part of these wonderful mathematical models that, for reasons no one really understands, provide us with very accurate predictions of how the world behaves. And it turns out that many of these very successful models use uncountable objects.

Martin Argerami
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The number of potential states of geometric frustration for all potential bosonic and fermionic particle interactions within the Hubble sphere since t=0 would be considered uncountable because the limit of this sum diverges. I don't know if that could be construed to include the multiverse model because t=0(Big Bang) to t=today(2016 CE) is finite.

Aaron Johnson
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  • Huh? $n=\sum_{i=1}^n 1$ diverges as $n\to\infty$ but that doesn't mean that the natural numbers are uncountable... – David Richerby Jun 12 '16 at 11:57
  • I am unfamiliar with using the math text on this site, and the character buffer limit won't allow me to post my original answer. On a Lebesgue vector space that describes all the possible degenerate particle interactions in the Hubble Sphere, the sum you give is absolutely convergent, not divergent https://en.wikipedia.org/wiki/Absolute_convergence The difference between a countably infinite series and a convergent infinite series is given here: https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_countably_infinite_dimensions_and_.E2.84.93.E2.80.89p_spaces. – Aaron Johnson Jun 16 '16 at 03:42
  • There's a guide to using $\LaTeX$ to typeset maths on Stack Exchange sites over on Computer Science meta. (There's probably one somewhere here, too, but I'm more familiar with the CS site.) – David Richerby Jun 16 '16 at 07:57
  • My point was that series convergence/divergence and (un)countability are two completely separate concepts and you can't deduce one from the other. In particular, "This sum diverges" does not imply "it is uncountable", as my example with the natural numbers shows. – David Richerby Jun 16 '16 at 08:00
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I'm no physics expert, but I would say the amount of frequencies in a ray of light, or the amount of frequencies in a sound wave, to name a couple. These are rendered countable when discretized, as computers do to store the information. The naturally occuring phenomena are continuous, though, and I would say they contain an uncountably infinite amount of "parts".

Juan Carlos Coto
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Aside from the fact that your definition of uncountability is wrong, as many others pointed out, I would say that there is something in our physical world that could be uncountable, and even have the cardinality of the continuum (i.e. the cardinality of the real numbers): I'm talking about spacetime.

In the framework of classical physics and general relativity, the number of points in spacetime is treated like a continuous set, that is to say a set with the same cardinality of the real numbers. But in some quantum-gravity models, spacetime becomes discrete at tiny scales: scales of the order of the Planck length, $l_P \simeq 1.62 \cdot 10^{-35}$ m.

Of course it is extremely difficult to probe the property of spacetime at such tiny scales, so we currently don't really know wether spacetime is continuous or not. But if it is, it would be an example of an uncountable entity in physics.

See also this question and this article.

Community
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valerio
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Many things in mathematics and physics can effectively be done with countable models. One thing were countable models might be insufficient is game theory with probability and independence. Say one player uses a probabilistic strategy involving a true source of randomness, and even his most powerful and omniscient opponent is unable to predict the future outcome of the source of randomness (even so he might be able to know the probabilistic strategy used by the player).

The intuition here is that independence might be a property which needs enormous amounts of space. Maybe one doesn't even need randomness for this, the phenomenon of independence already occurs for Henkin quantifiers without any randomness. Note however that I didn't even (try to) prove in any meaningful way that uncountability is required for stochastic independence.

If you believe that uncountability simply doesn't exist in the physical world, then you can just declare that randomness and independence are only apparent phenomena with no truly physical basis, like Max Tegmark did for his Mathematical Universe Hypothesis.

Thomas Klimpel
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