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 ß ^-collection axioms to any bounded first-order theory R of arithmetic yields an extension which is V ̂ -conservative over R. Preliminaries. A theory of arithmetic R contains the nonlogic symbols 0, S, +, ■, and < . 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) < is a linear ordering. (b) For every term t(x), there is a term a, such that R h xx <>>, a • • • Axk<yk-* t(x) < o,{y). Of course, the usual bounded theories of arithmetic, for example 7A0 or S\, are sufficient. Indeed letting a, be t suffices for these theories. Although Theorem 1 below holds for second order bounded theories of arithmetic such as U{ and V{, we shall only discuss first order theories in this paper. From now on, R is presesumed to be a first order theory. The syntax of first order logic is enlarged to include bounded quantifiers of the forms (Vx < t) and (3x < t), where / is any term not containing x. In [1] it is shown how Gentzen's sequent calculus LK may be enlarged to incorporate bounded quantifiers. A formula is bounded if and only if it contains no unbounded (i.e., usual) quantifiers. A theory R of arithmetic is bounded if and only if R is axiomatized by a set of bounded formulae. The class of 2°-formulae is defined to contain those formulae in which each unbounded quantifier is either existential and in the scope of an even number of negations, or universal and in the scope of an odd number of negations. Note that our definition of 2° is slightly broader than the set of 2j formulae defined by Paris and Kirby [4]. The B2,¡-collection axioms are (Vx < a)(3y)A(x,y) -» (3z)(Vjc < a)(3y < z)A(x,y), where A is any Sf-formula [4]. Note that A may contain additional free variables as parameters. The ¿?2°-collection axioms are equivalent to the 52°-collection axioms of Paris and Kirby [4] since the B2^-collection can prove that every 2° 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.