Let's suppose our computation model allows parallel computations
(Countably) infinite unions of semidecidable sets are semidecidable
To semidecide
: In parallel, try to semidecide each of the Ai
If any is true, one of the semidecisions will succeed in finite time
When it does, halt and say "YES"
(Warning: There are some important qualifications here that I will skip over.)
(Computer scientists must be much less cavalier than mathematicians about infinite operations.)
(e.g., what if i ranges over a noncomputable subset of ℕ?)