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

128 a potential subtlety concerning the distinctionnecessarily Output(π(e 0 ):M) ≠∅and so the sequence 〈(p i ,s i ,n i ):i ≤ b〉 mustboth exist and have the property that s b = s. For each i ≤ b, let N i be such thatN i is the size of p i as calculated in M. We first show that N b /∈ N, (i.e., N b isnonstandard). Assume toward a contradiction that N b is standard. Therefore, p bis an L + 0-proof witnessingPA ∪ 0 1 (M; s b) ⊢ “Output(e 0 :N) ≠ c”But we have that the following hold given that s = s b :(4.1) (M; s) PA ∪ 1 0(M; s b),(4.2) Output(e 0 :M) = Output(π(e 0 ):M) = s b ;hence, this is a contradiction. Therefore, N b is nonstandard. If T is not consistent,then there exists a proof, p, witnessing(M; t) ∪ 0 1 (M; t) ⊢ “Output(e 0:N) ≠ c”and this proof p necessarily has size strictly less than N b . But then within M it ispossible to define (p b+1 ,s b+1 ,n b+1 ) because one can choose p b+1 to have size atmost the size of p and thereby satisfy the requirement that p b+1 have size strictlyless than the size of p b , and this again is a contradiction.Therefore, T is a consistent theory, and so by Theorem 2, becauseand thatT ∈ SS(M)T = PA ∪ 0 1 (M; t) ∪ φ 0(c)where φ 0 (x) is the formula of L 0 , “Output(e 0 :N) = x,” there exists N such that(5.1) N PA,(5.2) M is an initial segment of N ,(5.3) N φ 0 [t].But then Output(e 0 :N ) = t, and so N is as required.The proof of Theorem 5 easily adapts to prove the following more general version.Theorem 6. Suppose that T is a recursive L 0 -theory such that PA ⊆ T . Thereexists e T ∈ N such that for all countable models,M Tif s = Output(e T :M) and if t is an internal binary sequence of M such that sis a proper initial segment of t, then there exists a countable model N T suchthat M is a proper initial segment of N and such thatOutput(e T :N ) = t.