Hierarchies of subsystems of weak arithmetic
Shahram Mohsenipour · Cambridge University Press eBooks · 2011
. We completely characterize the logical hierarchy of various subsystems of weak arithmetic, namely: ZR, ZR + N, ZR + GCD, ZR + Bez, OI + N, OI + GCD, OI + Bez. § 1. Introduction. In 1964 Shepherdson [6] introduced a weak system of arithmetic, Open Induction (OI), in which the Tennenbaum phenomenon does not hold. More precisely, if we restrict induction just to open formulas (with parameters), then we have a recursive nonstandard model. Since then several authors have studied Open Induction and its related fragments of arithmetic. For instance, since Open Induction is too weak to prove many true statements of number theory (It cannot even prove the irrationality of √2), a number of algebraic first order properties have been suggested to be added to OI in order to obtain closer systems to number theory. These properties include: Normality [9] (abbreviated by N), having the GCD property [8], being a Bezout domain [3, 8] (abbreviated by Bez), and so on. We mention that GCD is stronger than N, Bez is stronger than GCD and Bez is weaker than IE 1 ( IE 1 is the fragment of arithmetic based on the induction scheme for bounded existential formulas and by a result of Wilmers [11], does not have a recursive nonstandard model). Boughattas in [1, 2] studied the non-finite axiomatizability problem and established several new results, including: (1) OI is not finitely axiomatizable, (2) OI + N is not finitely axiomatizable.