User:IssaRice/Computability and logic/Characterization of recursively enumerable sets

Let $S$ be a set of natural numbers. The following are all equivalent.

Property	Emphasis
The elements of $S$ can be enumerated in a computable manner as $a_0,a_1,a_2,\dots$	Recursively enumerable
$S$ is the range of a computable partial function.	Recursively enumerable
$S$ is empty or the range of a computable total function.	Recursively enumerable
$S$ is empty or the range of a primitive recursive function.	Recursively enumerable
$S$ is the domain of a computable partial function.	"Semi"-ness
A recursive semicharacteristic function for $S$ exists.	"Semi"-ness
There exists a two-place recursive relation $R$ such that $x\in S⟺\mathrm{\exists }tR(x,t)$	"Semi"-ness
The relation $x\in S$ is ${\mathrm{\Sigma }}_1$	"Semi"-ness
The relation $x\in S$ has a computable partial verifying function, i.e., there exists a computable partial function $f$ such that $x\in S⟺f(x)\simeq 1$.	"Semi"-ness