all arithmetical subsets of ω and relations on ω belong to l (because the arithmetic definition gives one in l). conversely, any subset of ω belonging to l is arithmetical (because elements of l can be coded by natural numbers in such a way that ∈ is definable, i.e., arithmetic). on the other hand, l already contains certain non-arithmetical subsets of ω, such as the set of (natural numbers coding) true arithmetical statements (this can be defined from l so it is in l).