Cantor’s diagonal proof revisited In his diagonal argument (although I believe he originally presented another proof to the same end) Cantor allows himself to manipulate the number he is checking for (as opposed to check for a fixed number such as $\pi$), and I wonder if that involves some meta-mathematical issues. 
Let me similarly check whether a number I define is among the natural numbers. The number is $n+1$ and it is clear that $1$, $2$, $3,\ldots,\ n$ are not among these numbers. This “proves” that $n+1$ is not a natural number.
I have here, just like Cantor, a formula for a number, rather than a given number. What is the difference between our proofs?
It seems to me that the answer is that Cantor’s number (as opposed to mine) is being successively better bounded; the process of moving forward in his enumeration describes a converging series, such as we use to define the real numbers, using Cauchy limits. You don’t hear this added comment in the proof. Don’t you think it belongs to the proof? 

In the diagonal argument, a function $f$ from the set of sequences of real numbers to $\mathbb{R}$ is defined. We start from any sequence $S$ of real numebrs. Then it is shown that $f(S)$ is not an element in $S$. The formula is not "changing during the process"; the number we are searching for, $f(S)$, is well-defined if $S$ is given. Now we have shown that for any sequence $S$ of real numbers, there is a real number which is an element of $S$.
The $n+1$ in your proof is not a definition of a number you are searching.
