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There are a lot of random variables in physics that are, in principle, continuous and unbounded (canonical examples: time, position along each direction). In practice, it is my understanding that the number of outcomes of every possible experiment will be a finite integer. That is, the data is always binned, even if only implicitly, at some minimum scale and finite in scope.

Reading the length of something off of a ruler, for example, will always produce a number that is rounded to some nearest rational number, whether that number is a line on the ruler or interpolated from the scale by eye. It will also always only produces either a finite length answer or a lower limit, assuming the experiment has to be carried out in finite time (i.e. using a single ruler you can only translate it a finite number of times to measure something longer than it).

In principle, though, the length of something being measured by a ruler can assume a continuum of values, but this question isn't about the theoretical principle, it's about experiments that are realizable in finite time using finite resources. Is there any example of a real experiment that has an actual infinite number of outcomes in any way? If not, is there a good reason to think that we will never be able to produce such an experiment/measurement?

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  • $\begingroup$ There are many of cases in which the number of outcomes is not practically "countable", depending upon what you mean by that word...? $\endgroup$ – JMLCarter Jun 13 '18 at 22:34
  • $\begingroup$ Countable = finite integer. $\endgroup$ – Sean E. Lake Jun 13 '18 at 22:39
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    $\begingroup$ When you measure the distance between two ticks on a metal bar you are going to record one of a countable number of results, but that does not mean that there are only a countable number of distance between the marks: it means that the measuring procedure is of limited precision. $\endgroup$ – dmckee Jun 13 '18 at 22:55
  • $\begingroup$ @dmckee Exactly. Is there an example of an experiment where the infinite possibilities are fully exposed to the experimenter, or is it always finite precision/binned in some way? If so, why? $\endgroup$ – Sean E. Lake Jun 13 '18 at 22:57
  • $\begingroup$ The infinite number of possibilities are exposed to the experimenter when measuring the distance between two marks. It's not the fault of that distance that the experimenter is finite and using finite tools. $\endgroup$ – dmckee Jun 13 '18 at 23:01
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Even if we manage to construct such an experiment, we would not be able to write down the outcome.

To write down our outcome, we would need to encode it as a string. This string would be of finite length, taken from an alphabet of a finite number of symbols (i.e. A-Z). Such a string can describe a countably infinite number of possible outcomes. So right away, we can discount the possibility of experiments whose results come from an uncountably infinite set of possibilities.

Practically speaking, however, all real mediums of communication require the size of the string to be bounded. If you have 40 pages in a journal article to describe your results, you only have 120000 characters to capture your result (assuming roughly 3000 characters per page). This can only describe $26^{120000}$ possible outcomes, assuming the alphabet from A-Z. If there are 3 trillion trees on earth, and 20,000 pages can get produced from each tree, and you permit all of unicode ($2^{16}$ characters) rather than just A-Z, and you use all of them up for your paper you can get $2^{960,000,000,000,000,000}$ possible outcomes, which is gargantuan, but still a far cry from infinity.

It can be shown that you can develop a string to encode any real number, while still being able to do all integer arithmetic without violating any rules. However, you can't encode every real number at the same time with this trick. You have to pick the real number, and then build the system around it. In science, constructing your measurement scheme around the result after the result is acquired is considered to be poor form.

Accordingly, all physically realizable experiments map any countably infinite values into a finite space, if for no other reason than to provide a way to actually write the results down. There are, of course, plenty of other reasons to do this mapping, but this particular one is the easiest to argue from a information theory perspective.

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  • $\begingroup$ A really lovely answer! $\endgroup$ – tfb Jun 13 '18 at 23:17
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All practical physical measurements have a precision associated with them (this is distinct form uncertainty or accuracy). Like the rounding error on your ruler; this limits the number of outcomes to the $$\frac{\text{Measureable range}}{\text{Precision}} \, .$$

Additionally it is not possible to actually measure an infinite range of results. That's because there is no practical means by which a truely infinte magnitude could be recorded. (Not enough atoms in the universe.)

Consequently we can be sure that we do not have the means to make measurements that are able to return an infinite number of outcomes.

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