An uncountable set is a set that isn’t countable. It’s a bit more complicated than simply deciding whether a set contains a countable number of items (or not).
Infinity and The Uncountable Set
While you can’t imagine counting an infinite set (i.e., start at zero and count to infinity), an infinite set actually considered countable under one specific circumstance: when it is equal to the set of natural numbers (the counting numbers). That’s because there is a defined and explicit way to count the set (e.g. start at 1 and count upwards to include every whole number), even if you keep on counting forever. What matters here is that there is a defined rule by which you could (theoretically at least) count the items. Another way to say this: an infinite countable set can be put in one-to-one correspondence with positive integers.
All other infinite sets, except for the one exception above are uncountable (Li, 1999). For example, the set of all real numbers is uncountable. Note: There is a way count these sets, but in order to do that, you need a different type of infinity—one that is bigger than countable infinity (Pyke, 2007). So the “uncountable” here is a bit of a misnomer and really only refers to the fact that it’s “uncountable” in the sense of the natural numbers.
The Uncountable Set in Probability?
In probability, it’s important to be able to recognize an uncountable set because many axioms and properties of probability do not apply to infinite sets. For example, countable additivity (the probability of a union of a collection of disjoint events is the sum of their individual probabilities) does not apply.
References
Bal Gupta, S. (2014). Discrete Structures. Laxmi Publications Pvt Ltd.
Li, X. (1999). Probability, Random Signals, and Statistics. CRC Press.
Pyke, R. (2007). Weekly commentary: MAT335 – Chaos, Fractals and Dynamics. Retrieved November 18, 2020 from: https://www.sfu.ca/~rpyke/335/W00/7jan.html