Proving that $\{x|\varphi_x \; \text{is extendible to a total computable function} \} \neq \omega$ The problem that I'm working on is to prove that 
$$ Ext=\{ x \ | \ \varphi_x \text{is extendible to a total computable function}\} $$ 
is not equal to $\omega$. Here $\varphi_x$ is the $x$-th partial computable function. I'm having problem understanding what $Ext$ is. Does it mean that if $x \in Ext$ then $x$  is in some $W_i \subset W_j$ where $W_j$ is computable? But doesn't that mean $W_i$ is also computable?
More on the context of this problem, this is part 2 of a problem from Soare's book. Part 1 is to prove that there exists computably inseparable r.e. set, which I did.

A variant: Let $f(i)$ be the number of steps Turing machine number $i$ takes to stop on an empty tape, or undefined if Turing machine number $i$ doesn't stop. That is clearly a partial computable function. But there can be no total computable function $g$ that extends $f$ (that is, such that $g(i)=f(i)$ whenever $f(i)$ is defined). Namely, if such a $g$ existed, we could decide the halting problem simply by running a given machine for $g(i)$ steps and see whether it stops by then or not.
Therefore the index of $f$ in the sequence of all partial computable functions is not a member of $\mathrm{Ext}$.
