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 $$\mathbb{F}$$ be the set of all functions from $$\N$$ to $$\N$$.

Clearly $$\mathbb{F}$$ 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 $$\Phi: \N \to \mathbb{F}$$ 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 $$g \in \mathbb{F}$$, 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 $$\mathbb{F}$$ 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 $$\mathbb{F}$$ and $$\N$$ and so $$\mathbb{F}$$ is not equivalent to $$\N$$.

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