Uncountability (Math 3)

A $X$ is uncountable if there is no between $X$ and . Equivalently, there is no from $X$ to $N$.

Foundational Considerations

In set theories without the , such as without choice (ZF), it can be that there is a $\kappa$ 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 , so it doesn't make sense to think of uncountability as "larger" than ${\aleph }_0$. In the presence of choice, , so an uncountable set can be thought of as "larger" than a countable set.

Countability in one is not necessarily countability in another. By , there is a model of set theory $M$ where its of the naturals, denoted $2_M^N\in M$ is countable when considered outside the model. Of course, it is a that $2_M^N$ is uncountable, but that is within the model. That is, there is a bijection $f:N\to 2_M^N$ 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 .

See also

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