On the Strength of the Interpretation Method

In spite of the fact that true arithmetic reduces to the monadic second-order theory of the real line, Peano arithmetic cannot be interpreted in the monadic second-order theory of the real line. ?0. Introduction. The decision problem for the monadic second-order theory of the real line was posed by Grzegorczyk in 1951 [Gr], and was proved undecidable in 1976 by Shelah [Sh]. Shelah reduced the first-order theory of true arithmetic to the monadic second-order theory of the real line. In other words, he constructed an algorithm that, given a sentence p in the first-order language of arithmetic, produces a sentence p' in the monadic second-order language of order such that p is true in the standard model of arithmetic if and only if p' is true in the real line. Shelah's proof was analyzed, strengthened and generalized in several papers by the present authors (see [Gu2]). In particular, the following somewhat strange results were proved in [GS]: Let V be a model of ZFC, B the Boolean algebra of regular open subsets of the real line in V, and VB the corresponding Boolean-valued model of ZFC. The full second-order VB-theory of arithmetic reduces to the monadic second-order Vtheory of the real line. If V satisfies the continuum hypothesis then the full secondorder VB-theory of the real line reduces to the monadic second-order V-theory of the real line. In spite of these strong reducibility results, we prove in this paper that Peano arithmetic is not interpretable in the monadic second-order theory of the real line. Actually, we prove a stronger result which is also more convenient to prove. To formulate the stronger result we need a couple of definitions. DEFINITION 0.1. We define a first-order theory with equality which will be called the weak set theory. The signature of the weak set theory consists of one binary predicate symbol P, and the axioms of the weak set theory are as follows: (a) Vx3yVz[P(z, y) +-* z = x]. (b) VxVy3uVz[P(z, u) +-+ (P(z, x) or P(z, y))]. (c) 3xVy1-P(y, x)]. Received August 25, 1986; revised September 29, 1987. These results were obtained in principle during the 1980-81 academic year when both authors were fellows in the Institute for Advanced Studies of The Hebrew University of Jerusalem. The final preparation of the paper was supported in part by NSF grant DCR 85-03275 and by a grant from the U.S.-Israel Binational Science Foundation. ? 1989, Association for Symbolic Logic 0022-4812/89/5402-0001 /$02.90