An Introduction to Basic Arithmetic

Logic Journal of the IGPL 16 (1):1-13 (2008)
  Copy   BIBTEX

Abstract

We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of , a fragment of PA with Induction restricted to the formulas made up of ∃, ∧ and/or ∨. We also study an interesting extension of BQC, called EBQC, which is the extension by the axiom schema ⊤ → ⊥ ⇒ ⊥. We show that this extension behaves very like to IQC, and the corresponding arithmetic, EBA shows similarities to HA.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,616

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Basic Predicate Calculus.Wim Ruitenburg - 1998 - Notre Dame Journal of Formal Logic 39 (1):18-46.
Comparing Peano arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
An Effective Conservation Result for Nonstandard Arithmetic.Erik Palmgren - 2000 - Mathematical Logic Quarterly 46 (1):17-24.
On the structure of kripke models of heyting arithmetic.Zoran Marković - 1993 - Mathematical Logic Quarterly 39 (1):531-538.
Classical and Intuitionistic Models of Arithmetic.Kai F. Wehmeier - 1996 - Notre Dame Journal of Formal Logic 37 (3):452-461.
Polynomially Bounded Recursive Realizability.Saeed Salehi - 2005 - Notre Dame Journal of Formal Logic 46 (4):407-417.
Unfolding finitist arithmetic.Solomon Feferman & Thomas Strahm - 2010 - Review of Symbolic Logic 3 (4):665-689.
Closed fragments of provability logics of constructive theories.Albert Visser - 2008 - Journal of Symbolic Logic 73 (3):1081-1096.

Analytics

Added to PP
2015-02-04

Downloads
18 (#711,820)

6 months
1 (#1,042,085)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Proof complexity of substructural logics.Raheleh Jalali - 2021 - Annals of Pure and Applied Logic 172 (7):102972.
The de Jongh property for Basic Arithmetic.Mohammad Ardeshir & S. Mojtaba Mojtahedi - 2014 - Archive for Mathematical Logic 53 (7-8):881-895.
A unification of the basic logics of Sambin and Visser.M. Ardeshir & V. Vaezian - 2012 - Logic Journal of the IGPL 20 (6):1202-1213.

View all 6 citations / Add more citations

References found in this work

No references found.

Add more references