enumeration theorem

there is a partial computable function of two variables $\text{[math]}$ such that $\text{[math]}$ . by turing's theorem there is some $\text{[math]}$ such that $\text{[math]}$ .
[cite:;taken from @computability_soare_2016 theorem 1.5.3]

we write
$\text{[math]}$ basically $\text{[math]}$ is a set of inputs that the computable function $\text{[math]}$ (which has the godel number $\text{[math]}$ ) converges on, and it denotes the $\text{[math]}$ th recursively enumerable set.
[cite:;taken from @computability_davis_1994 chapter 4 recursively enumerable set; theorem 4.6 enumeration theorem]

in [cite:@computability_soare_2016], the notation $\text{[math]}$ is used to denote the inputs for which the function whose number is $\text{[math]}$ halts on in less than $\text{[math]}$ steps.
[cite:;refer to @computability_soare_2016 definition 1.6.17]

a set $\text{[math]}$ is r.e. if and only if there is an $\text{[math]}$ for which $\text{[math]}$ .
[cite:;taken from @computability_davis_1994 theorem 4.6]