An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

Journal of Symbolic Logic 70 (4):1171-1209 (2005)
This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae. Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions
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DOI 10.2178/jsl/1129642122
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References found in this work BETA
On the Scheme of Induction for Bounded Arithmetic Formulas.A. J. Wilkie & J. B. Paris - 1987 - Annals of Pure and Applied Logic 35 (3):261-302.
On Herbrand Consistency in Weak Arithmetic.Zofia Adamowicz & Paweł Zbierski - 2001 - Archive for Mathematical Logic 40 (6):399-413.
Derivability Conditions on Rosser's Provability Predicates.Toshiyasu Arai - 1990 - Notre Dame Journal of Formal Logic 31 (4):487-497.
Subsystems of True Arithmetic and Hierarchies of Functions.Z. Ratajczyk - 1993 - Annals of Pure and Applied Logic 64 (2):95-152.

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Citations of this work BETA
Passive Induction and a Solution to a Paris–Wilkie Open Question.Dan E. Willard - 2007 - Annals of Pure and Applied Logic 146 (2):124-149.
2006 Annual Meeting of the Association for Symbolic Logic.Matthew Valeriote - 2007 - Bulletin of Symbolic Logic 13 (1):120-145.

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