Forcing and models of arithmetic

It is shown that every countable model of Peano arithmetic with finitely many extra predicates (or of ZFC with finitely many extra predicates) is a reduct of a pointwise definable such model. This note applies the forcing method to a question concerning definability in models of Peano arithmetic. THEOREM. Let M= (IMI, +, ) be a countable model of Peano arithmetic.2 Then there is a set Uc IMI such that (i) (M, U) satisfies the first order induction schema for formulas containing an extra predicate U(x); (ii) every element of IMI isfirst order definable in (M, U). PROOF. A condition is an M-finite sequence of O's and l's, i.e. a mapping p:{bIb<Ma}-+{O, 1} such that a E IMI and p is coded by an element of IMI. We use p, q, . . . as variables ranging over conditions. A set of conditions is dense if every condition is extended by some condition in the set. Let (anIn<w)) enumerate the elements of IMI. Let (DnIn<wo) enumerate the dense sets of conditions which are first order definable over M allowing parameters from IMI. It is safe to assume: the parameters in the first order definition of Dn are among ao, a,, * * , an-l Define a sequence of conditions (pnIn<wo) by po= 0; p2n+?= the <3m least condition q 2P2n such that q E Dn; P2n+2 =P2n+1 followed by a string of an O's followed by a 1. Define U c IMI by letting U {pnIn < co} be the characteristic function of U. To prove that (M, U) satisfies first order induction, use the genericity of U. [Details. Let L be the first order language with +,, U(x), and constant symbols a for each a E IMI. For 0 a sentence of L define the (strong) Received by the editors May 16, 1973. AMS (MOS) subject classifications (1970). Primary 02H05, 02H13, 02H20. 1 This research was partially supported by NSF contract GP-24352. 2 Peano arithmetic is the theory P of Shoenfield, Mathematical logic, AddisonWeplev 1 967 ? American Mathematical Society 1974