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

the program e 0 121of the nonnegative integers with the operations of addition and multiplication and withthe associated total order. We adopt as our base theory (i.e., the specified set of “physicallaws”) the standard Peano axioms as implemented in formal logic; these are the formalaxioms, PA. In fact, all of our considerations apply to any (recursive) extension of theseaxioms.We restrict consideration to the collection of Turing programs that, by the formatof the program, generate a finite binary sequence in a specific fashion: the programattempts to compute a finite sequence,〈(s i ,N i ):i ≤ m〉by successively computing each pair (s i ,N i ), such that for all i + 1 ≤ m, N i+1 >N i ,s i ,s i+1 are each (nonempty) binary sequences, and s i is an initial segment of s i+1 .If the program fails to compute such a sequence – in particular, if the program failsto compute (s 0 ,N 0 ) – then there is no output generated; otherwise, the sequence s mis the (total) output of the program. Increasing the running time of the program canonly alter the output by lengthening the output sequence (i.e., by creating additionaloutput), but this can only happen if N m > 0. Thus, once the program succeeds incalculating (s 0 ,N 0 ), the output of the program can only increase at most N 0 timesas the running time increases. Our convention is that for such Turing programs e, ifno output is generated, then the output is empty. This is unambiguous because, bythe format we have described, the program can never generate the empty sequence asoutput (s m ≠∅). We could just as easily require, by altering the format, that outputis always produced, but the technical details would be less natural. The point is thatwhatever the format, the output generated in the standard universe must be an initialsegment of the output generated in any possible nonstandard universe, and it is onlythe potential for additional output that we are interested in.By combining the construction of a Gödel sentence with the Kleene recursiontheorem and appealing to the Friedman isomorphism theorem, we construct an indexe 0 for a Turing program in the format just described and with the following property.For any countable model M PA, if t is any (internally) finite binary sequence of Mthat properly extends the sequence that is the output of the program e 0 as implementedwithin M (if any such output exists), then there is a countable model, N PA, suchthat(1) M is an initial segment of N ,(2) t is exactly the (total) output of the program e 0 generated within N .Any index e 0 satisfying this condition (and in the format indicated above) clearlywitnesses the informal claims we have made, and e 0 easily gives the existence of aprogram e0 ∗ as claimed. We are now faced with a rather precise mathematical problem(to find e 0 with the indicated property).For the construction of e 0 we are assuming that PA is formally consistent, and forthe specific e 0 we construct, ifM PA