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So I was reading a a question and top comment on math stack exchange that didn't make sense to me.

you can measure the temperature of something, but you can't count it. Incidentally, I claim cardinality is not the relevant feature: I would describe a random variable taking values in, say, the set of all subsets of $N$ as still discrete, even though the set of all subsets of $N$ is uncountable!

where discrete refers to:

discrete: (of a variable or data) assuming a value from a finite or countably infinite sample space;

It isn't clear to me what this means? I'd love the physical intuition or at least an example behind this?

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    $\begingroup$ I think you're confused because you think counting is the same as measurement, right? But actually, counting is an act of assigning natural numbers to a sequence of objects. One, two, three, ... $\endgroup$
    – Ruslan
    Commented Jun 1, 2023 at 5:07
  • $\begingroup$ Yes but i can assign each measurement a natural number too right? Would love if you could elaborate? $\endgroup$ Commented Jun 1, 2023 at 5:09
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    $\begingroup$ This would count your measurements, not the quantity you measure. I.e. you'll know how many measurements you've made, but not how hot the object being measured was. $\endgroup$
    – Ruslan
    Commented Jun 1, 2023 at 5:10
  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – Buzz
    Commented Jun 1, 2023 at 21:57

3 Answers 3

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In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set contains 3 elements, and therefore. has a cardinality of 3.

Temperature T is an average quantity characterizing the total of the particles in the sample. For usual set the cardinality of the sample with a temperature T is enormous, it will only play a role in the statistical errors in measuring T.( the more particles in the sample the better the accuracy).

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I think you have misunderstood the comment to the original question. It is not saying that temperature is a discrete attribute. It is saying that temperature is an example of an attribute that can be measured but is not discrete, and so cannot be counted. In classical physics many attributes can be measured but are not discrete, since they have no smallest increment. Other examples of continuous attributes in classical physics include mass, time and length.

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  • $\begingroup$ But that fundamentally misunderstands physics as well. A "quantity" in physics is the comparison between two physical systems. It's not a number that has mathematical countability properties. When we are measuring 1.7m in length, what we actually mean is that we have a physical system that has a reasonably stable length that we call "1m". We then compare that to our system for which we want to know the length and find that it's longer, but not quite twice as long. We iterate with fractions for a bit until we run into measurement errors. Every result within those errors is equivalent. $\endgroup$ Commented Jun 1, 2023 at 18:51
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Physical variables are almost never discrete, but rather continuous (although some discontinuities in Physical systems may exist). Same for the temperature,- it's from the real number domain $T \in [0; T_P] \approx [0; 10^{32}K]$. So countability is not applicable here.

On the contrary, quantum numbers are discrete sets of (half-)integers which can be countable.

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