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Hierarchies of subsystems of weak arithmetic

Published online by Cambridge University Press:  07 October 2011

By
Shahram Mohsenipour
Edited by
Juliette Kennedy  and
Roman Kossak
Juliette Kennedy
Affiliation:
University of Helsinki
Roman Kossak
Affiliation:
City University of New York
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Summary

Abstract. 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 IE1 (IE1 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.

Type
Chapter
Information
Set Theory, Arithmetic, and Foundations of Mathematics
Theorems, Philosophies
 , pp. 80 - 92
Publisher: Cambridge University Press
Print publication year: 2011

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References

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