An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
Dan E. Willard · Journal of Symbolic Logic · 2005
Abstract 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 systemD provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2)Ddoes not allow for the use of a modus ponens methodology above essentially the levels of Π1and Σ1formulae. 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.