Enumerability and diagonalization

Question

I'm taking a course dealing with Boolos' "Computability and Logic" and I'm stuck on a particular homework assignment. Where $\mathbb Z_{>0}$ is the set of positive integers, prove that the set of all one-to-one, total functions from $\mathbb Z_{>0}$ into $\mathbb Z_{>0}$ is not enumerable.

Knowing that 1-1 means not repeating, into means that some elements may be missing, and a total function means no undefined elements. How would I approach this?

Answer 1

HINT: Make it strictly increasing and make sure that if your functions are $\{f_n:n\in\Bbb N\}$, then $f(n)>f_n(n)$ for each $n\in\Bbb N$. I’ve spoiler-protected one way to do this.

Let $$f(n)=1+\max\big(\{f(k):k<n\}\cup\{f_n(n)\}\big)$$ for each $n\in\Bbb N$.