Existence of Uncomputable Mappings

Theorem
There exists a function that is not computable.

Proof
Every Turing Machine can be Represented as a Binary String.

Let $M_\alpha$ be the Turing machine represented by the binary string $\alpha \in \{ 0,1 \}$. Consider the function:


 * $ UC(\alpha) \ = \begin{cases}

1 & if \ M_\alpha(\alpha) = 0 \\ 1 & if \ M_\alpha(\alpha) \ does\ not\ halt \\ 0 & if \ M_\alpha(\alpha) = 1 \end{cases}$

Assume for contradiction $UC$ is computable on a Turing Machine.

By the assumed hypothesis there is some machine that computes $UC$. If $M_{UC}$ is the binary string that corresponds to that machine then:


 * $M_{UC}(UC) = 1 \iff M_{UC}(UC) \ne 1$

Because this is obviously a contradiction we can conclude that $UC$ is not computable.