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). T

While you can’t imagine counting an infinite set, it’s 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 sat 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

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