A conservation result concerning bounded theories and the collection axiom

We present two proofs, one proof-theoretic and one model-theoretic, showing that adding the BX?-collection axioms to any bounded first-order theory R of arithmetic yields an extension which is VE:-conservative over R. Preliminaries. A theory of arithmetic R contains the nonlogic symbols 0, S, +, , and S . R may contain further nonlogical symbols; in particular S2 is a theory of arithmetic [1]. We shall say that R is sufficient if and only if R proves (a) (3z)('Vx < a)(3y < z)A(x, y), where A is any p0-formula [4]. Note that A may contain additional free variables as parameters. The BEl-collection axioms are equivalent to the BE-collection axioms of Paris and Kirby [4] since the B2r-collection can prove that every :1 formula is Received by the editors November 18, 1985 and, in revised form, February 21, 1986, 1980 Mathematics Subject Classification (1985 Revision). Primary 03C30, 03B99.