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

126 a potential subtlety concerning the distinctionwe identify the internal binary sequences of M with their corresponding canonicalcodes. Thus, for each internal binary sequence s of M, we interpret 1 0 (M; s)as the set of sentences φ(c)ofL + 0 such that φ(x)isa0 1 -formula of L 0 and suchthatM φ[a]where a is the canonical code of s. More generally, for any formula φ(x) ofL 0 ,we write(M,s) φ(c)to indicate that M φ[a], where again a is the canonical code of s.With this convention, for all L 0 -structures, M, such that M PA, for alle ∈ N, and for all internal finite binary sequences s of M (including the emptysequence), the following are equivalent:(1.1) Output(e:M) = s.(1.2) (M; s) “Output(e:N) = c.”Given our fixed Gödel numbering of the symbols and formulas of L + 0 , a formalproof p from L + 0is a finite sequencep =〈n i : i ≤ m〉where each n i ∈ N. We define the size of p to be n · m, where n =max {n i | i ≤ m}. There is a natural order on triples, (p, s, n), where p is aformal L 0 -proof of size k for some k ≤ n, s is a binary sequence of length k forsome k ≤ n, and n ∈ N:(p ∗ ,s ∗ ,n ∗ ) ≤ (p, s, n)if n ∗