Uncountability (Math 3)

A set \(X\) is uncountable if there is no bijection between \(X\) and \(\mathbb{N}\). Equivalently, there is no injection from \(X\) to \(\mathbb{N}\).

Foundational Considerations

In set theories without the axiom of choice, such as Zermelo Frankel set theory without choice (ZF), it can be consistent that there is a cardinal number \(\kappa\) that is incomparable to \(\aleph_0\). That is, there is no injection from \(\kappa\) to \(\aleph_0\) nor from \(\aleph_0\) to \(\kappa\). In this case, cardinality is not a total order, so it doesn’t make sense to think of uncountability as “larger” than \(\aleph_0\). In the presence of choice, cardinality is a total order, so an uncountable set can be thought of as “larger” than a countable set.

Countability in one model is not necessarily countability in another. By Skolem’s Paradox, there is a model of set theory \(M\) where its power set of the naturals, denoted \(2^\mathbb N_M \in M\) is countable when considered outside the model. Of course, it is a theorem that \(2^\mathbb N _M\) is uncountable, but that is within the model. That is, there is a bijection \(f : \mathbb N \to 2^\mathbb N_M\) that is not inside the model \(M\) (when \(f\) is considered as a set, its graph), and there is no such bijection inside \(M\). This means that (un)countability is not absolute.

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  • Uncountability

    Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.