Published online by Cambridge University Press: 12 March 2014
We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.
In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:
A0 = {X ⊆ ω: X is arithmetic},
Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},
Aλ = ⋃α<λAα (λ limit),
RA = ⋃αAα.
We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:
L0 = ⊘,
Lα+1 = {X: X is (first-order) definable over Lα},
Lλ = ⋃α<λLα (λ limit),
L = ⋃αLα.
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