Natural Number Functions are Uncountable

Theorem
The set of all natural number one-variable functions $\left\{{f: \N \to \N}\right\}$ is uncountably infinite.

Proof
Let $\Bbb F$ be the set of all functions from $\N$ to $\N$.

Clearly $\Bbb F$ is infinite because it contains for each $k \in \N$ the constant function $f_k: \N \to \N: f_k \left({n}\right) = k$ and these are all different and (trivially) countably infinite in number.

Let $\Phi: \N \to \Bbb F$ be a function.

For each $n \in \N$ let $f_n: \N \to \N$ be the function $\Phi \left({n}\right)$.

Let us define $g: \N \to \N$ by:
 * $g \left({n}\right) = f_n \left({n}\right) + 1$.

Then $g \in \Bbb F$, but $\forall n \in \N$, $g \left({n}\right) \ne f_n \left({n}\right)$ and so $g \ne f_n$.

Since $g$ is an element of $\Bbb F$ which is different from all the values taken by $\Phi$, it follows that $\Phi$ is not a surjection and hence not a bijection.

Thus no bijection exists between $\Bbb F$ and $\N$ and so $\Bbb F$ is not equivalent to $\N$.

Thus from Countably Infinite Iff Equivalent to Natural Numbers, $\Bbb F$ is uncountable.