A Note on a Result of Buss concerning Bounded Theories and the Collection Scheme

Samuel Buss showed that, under certain circumstances, adding the collection scheme for bounded formulae to a bounded theory of arithmetic yields a ∀Σ1conservative extensions. We present a very simple model theoretic proof of a generalization of this result. The general form of the collection scheme in the language of first-order Peano Arithmetic is as follows: ∀u ≤ x ∃y A(u, y) → ∃w ∀u ≤ x ∃ y ≤ w A(u, y) , where A is a formula that may contain additional free variables as parameters. This scheme is obviously true in the standard model and, indeed, it is provable in Peano Arithmetic for any formula A of the language. An early result of Charles Parsons (in [Ps70]) states that the theory I∆0 does not prove the collection scheme for bounded formulae (i.e., for formulae A that contain only bounded quantifiers: the so-called ∆0 or bounded formulae). Recall that the theory I∆0 is Robinson’s arithmetic Q together with the scheme A(0) ∧ ∀x (A(x)→ A(x + 1))→ ∀x A(x) , where A ∈ ∆0 (parameters are allowed). Nevertheless, Jeff Paris showed in [Pr80] that adding the bounded collection scheme to the theory I∆0 does not enable the deduction of new ∀Σ1-sentences. Paris’ proof hinges on the following result: Received : March 18, 1994; Revised : January 4, 1995. * This work was partially supported by project 6E92 of CMAF (Universidade de Lisboa).