Hello everyone just like the title says I want to prove that $H(x) = \mu y T(x,x,y)$ has no total computable extension such that if we had a function $BIG(x)$ that is both total and agrees with $H(x)$ whenever $H(x)$ is defined, then $BIG(x)$ is not computable. This is a homework question so I don't want a full solution just some help!
$\bf{NOTE}:$ The predicate $T(y,x,z)$ means that it holds iff program $y$ takes an input $x$ (could be $n$-ary) and has a computational history z! This is supposed to be the Kleene T predicate basically.
The function $H(x)$ I believe returns the smallest computational history $y$ such that a program $\{x\}(x)$ (program takes input of its own configuration and runs) runs and halts, since $\mu y R(x,y)$ means the smallest $y$ such that $R(x,y)$ holds. Maybe I am not quite clear what it means for $BIG$ to agree with $H(x)$ or what it's own input. I think I need to create a diagonal function that uses $BIG$ if $BIG$ was computable and show that if I had some program $e$ then it must agree with $BIG$ but based on my definition of that diagonal function it isn't. If you are reading this you might see the mess of my thinking, any help would be greatly appreciated!
