Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

the existence of e 0 125or Output(e:M) is a proper initial segment of Output(e:N ). We let Output(e : N) denotethe output of e generated within the standard model of PA (i.e., within our idealizeduniverse). Notice that with our convention, the assertion“Output(e:N) =∅”is simply the assertion that the program e generates no output. We also note that by ourconventions the assertion“Output(e:N) = s”is by complexity a 0 1 -assertion if s =∅and it is a 0 2-assertion if s ≠∅it is not ingeneral a 1 0 -assertion. However, the assertion“s is a proper initial segment of Output(e:N)”is easily verified to be a 1 0 -assertion. The next two theorems are versions of the Kleenerecursion theorem adapted to our setting (Kleene 1938).Theorem 3(Kleene recursion theorem). Suppose thatπ : N → Nis a recursive function. Then there exists e ∈ N such thatOutput(e:N) = Output(π(e):N).We shall need the uniform version of the Kleene recursion theorem.Theorem 4(uniform Kleene recursion theorem). Suppose thatπ : N → Nis a recursive function. Then there exists e ∈ N such thatOutput(e:M) = Output(π(e):M)for all L 0 -structures M such that M PA.We now come to our main theorem, which we can now precisely formulate usingthe notation we have fixed.Theorem 5.There exists e 0 ∈ N such that for all countable models,M PAif s = Output(e 0 :M), and if t is an internal binary sequence of M such that s isa proper initial segment of t, then there exists a countable model N PA suchthat M is a proper initial segment of N and such thatOutput(e 0 :N ) = t.proof.We shall abuse notation, and for any L 0 -structure M such thatM PA,