An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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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Walter Carnielli (2011). The Single-Minded Pursuit of Consistency and its Weakness. Studia Logica 97 (1):81 - 100.
Dan E. Willard (2007). Passive Induction and a Solution to a Paris–Wilkie Open Question. Annals of Pure and Applied Logic 146 (2):124-149.
Dan E. Willard (2006). A Generalization of the Second Incompleteness Theorem and Some Exceptions to It. Annals of Pure and Applied Logic 141 (3):472-496.
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