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

the existence of e 0 127(3.3) p i+1 has size k for some k ≤ n i+1 and the size of p i+1 is strictly less than thesize of p i ,(3.4) for each 1 0 formula φ(x)ofL 0 such that φ(c) occurs in p i+1 , the least witnessthat φ[s i+1 ] holds is below n i+1 .If (p 0 ,s 0 ,n 0 ) is not defined, then there is no output of e ∗ , and so with ourconventionOutput(e ∗ :N) =∅.Otherwise, Output(e ∗ :N) = s m , where m is largest such that (p m ,s m ,n m )isdefined. Viewing e ∗ as generating the sequence〈(s i ,N i ):i ≤ m〉,where for all i ≤ m, N i = size(p i ), clearly e ∗ can be required to have the properformat. Thus, there is a recursive functionπ : N → Nsuch that for each e ∈ N, π(e) is a Turing program that computes the binarysequence using e as described above.By the uniform Kleene recursion theorem, there exists e 0 ∈ N such that for allL 0 -structures, M,ifM PAthen Output(e 0 :M) = Output(π(e 0 ):M).We finish by proving that e 0 witnesses the theorem. Suppose that M is acountable L 0 -structure and thatM PA.Let s = Output(e 0 ; M), and let t be an internal binary sequence of M such thats is a proper initial segment of t.Let T = PA ∪ 1 0(M; t) ∪ φ 0(c), where φ 0 (x) is the formula of L 0 thatexpresses (in all models of PA)“Output(e 0 :N) = x.”Thus, T ∈ SS(M). We claim that T is consistent. We consider only the casethat Output(e 0 , M) ≠∅(i.e., the case that e 0 produces output in M); the casethatis similar (and simpler). LetOutput(e 0 , M) =∅〈(p i ,s i ,n i ):i ≤ b〉be the internal sequence of M generated by the program π(e 0 ) within M. BecauseOutput(e 0 :M) = Output(π(e 0 ):M)