The set of natural number functions is uncountable I thought the set of natural number functions would be of the same cardinality as the countably infinite product of $\mathbb{N}$, which is countable.
Each natural number function can be identified with an infinite-tuple of $\mathbb{N}$ by letting the $i$th entry be the image of the number $i$ under the function.

The cardinality of the natural number functions is $\mathbb{N}^{\mathbb{N}} \geq 2^{\mathbb{N}}$ and $2^{\mathbb{N}}$ is uncountable
