Computability and Logic

318 NONSTANDARD MODELS
for the construction of section 13.3. Call i minimal if there is no j < i such
that c i = c j is in Ɣ # . Show that the function δ taking n to the nth number i
such that c i is minimal is arithmetical.
25.10 Continuing the preceding problem, explain why for every constant c there
is a unique n such that c = δ(n) isinƔ # , and that if in the construction of
section 13.3 we take as the element ci
M associated with the constant c i this
number n, then the relation R M associated with any relation symbol R will
be arithmetical.
25.11 Explain how the arithmetical Löwenheim–Skolem theorem for the case where
function symbols are absent follows on putting together the preceding six
problems, and indicate how to extend the theorem to the case where they are
present.