User:IssaRice/Computability and logic/Characterization of recursively enumerable sets
Let be a set of natural numbers. The following are all equivalent.
|The elements of can be enumerated in a computable manner as||Recursively enumerable|
|is the range of a computable partial function.||Recursively enumerable|
|is empty or the range of a computable total function.||Recursively enumerable|
|is empty or the range of a primitive recursive function.||Recursively enumerable|
|is the domain of a computable partial function.||Semi-ness|
|A recursive semicharacteristic function for exists.||Semi-ness|
|There exists a two-place recursive relation such that||Semi-ness|
|The relation is , i.e., there exists a -place recursive relation such that||Semi-ness|
|The relation has a computable partial verifying function, i.e., there exists a computable partial function such that .||Semi-ness|
The rows labeled "Recursively enumerable" all emphasize the fact that is "recursively enumerable", i.e., that the elements of can be listed in a recursive (computable) way.
The rows labeled "Semi-ness" all emphasize the fact that is semidecidable/recognizable/verifiable/semirecursive, i.e., that if something is in , then in a finite amount of time we can verify this computably, but that if something isn't in , then we will run into an infinite loop.
The fundamental theorem of recursively enumerable sets says that recursively enumerable equals semi-ness.