Variations on the Collapsing Lemma

Consistency and Decidability in Some Paraconsistent Arithmetics

“…As a final note, in these, and every other kind of model, I'll fix that, for any variable assignment f, M f (x ≤ y) = M f (∃z(x + z = y)), so this will be left tacit from here on out. 7 I'll be interested in two kinds of finite collapse model here -on the left is a cyclic model with n elements (i.e. a cycle with period n), C n , and on the right is a heap model H m,n with a 'tail' of m − 1 initial elements, followed by a cycle of period n:…”

“…Among the most interesting of these have concerned inconsistent models of arithmetic. The investigation into such structures, starting in Meyer's work and further driven by Dunn's [6] remarkable result concerning three-valued model theory has led to a variety of interesting developments in inconsistent model theory; see [14,15,21,22,17,18,7]. Many of the noteworthy results are consequences of the fact that Peano Arithmetic PA, formulated in Priest's three-valued paraconsistent logic LP, has finite, and hence decidable, inconsistent models.…”