Let $f:\omega\to\omega$ be a bijective computable function. I am trying to prove that $f^{-1}$ is computable. I tried searching online this result, but I couldn't find any result that convinced me that there actually exists a Turing machine that can compute $f^{-1}$.
Can anyone please provide me a sketch of a construction of a Turing machine that computes $f^{-1}$?
I am a novice in the computability theory, and I am not yet fully convinced by Church-Turing thesis. So even though I know a human language algorithm to compute $f^{-1}$, I still wanted to know an explicit construction of a Turing machine that computes $f^{-1}$.
Here is my human language algorithm to compute $f^{-1}$: given an input $y$, search for $x$ such that $f(x)=y$ in the following way: Check whether $f(0)=y$, then $f(1)=y$, and so on. Then we will eventually find some $x$ such that $f(x)=y$, and we halt with the output $x$.