chapter  5
McAloon’s theorem
Pages 3

The importance of diagonal indiscemibles follows from the fact that from a suitable set of diagonal indiscemibles in a model of /Ao one can construct a model of PA. More precisely, let M be a model of / Ao, and let J be a nonempty set of diagonal indiscemibles for Ao-formulas with the further property that if c, d e J and c < d then c2 < d. Then Bj = { x : x < c for some c e J} is a model of PA. So, in order to prove McAloon’s result it is enough to realize the following recursive type r(/o, / i , . . . ) of Ao-formulas in countably many unknowns.