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

122 a potential subtlety concerning the distinctionis any model in which PA is consistent, there is no output of the program e 0 generatedwithin the model M. Therefore, if t is any finite binary sequence, there must exist amodelM PAin which the output of the program e 0 is exactly the given binary sequence t. This muchweaker property of e 0 corresponds to the theorem that the problem of determiningfor a given sequence whether the sequence is random on the basis of informationcontent is not decidable. Arguably, this weaker property alone does not illustrate anysubtlety in the distinction between determinism and nondeterminism in the physicaluniverse. The reason is that this weaker property of e 0 only holds for those idealizeduniverses (models of PA) in which an additional law holds (“PA is consistent”), andthis additional law is not on the specified list (PA). More precisely, the actual programe 0 that we produce (in the proof of Theorem 5) has the property that for any model,the following are equivalent:M PA,(1) M “PA is inconsistent.”(2) The program e 0 generates output within M.Finally, for any consistent, recursive theory T extending PA, there is an analogous indexe T that has the analogous property of e 0 but relative to models M T .5.3 The Existence of e 0The remainder of this chapter is simply concerned with the proof that e 0 exists, and wewill assume familiarity with the basic notions of formal mathematical logic. We beginby fixing some notation. We let L 0 denote the formal language for number theory; thisis the formal language with two binary function symbols (one for “+” and one for “·”)and a binary relation symbol (for “