Some properties of ∀∃ models in the isols
Thomas G. McLaughlin · Proceedings of the American Mathematical Society · 1986
It is a consequence of theorems proved by Nerode [ 10 ] and Hirschfeld [ 7 ] that every countable model of ∀ ∃ \forall \exists arithmetic is isomorphic to a subsemiring of a one-generator semiring of isols. We characterize, in terms of the generators of "Nerode semirings", the contents of arbitrary semirings R {\mathbf {R}} of isols that are models of ∀ ∃ \forall \exists arithmetic, and we show that all such R {\mathbf {R}} are in fact models of the ω \omega -true ∀ ∃ \forall \exists sentences of isol theory . We solve one of the chief problems left open in [ 8 ], and in § 3 \S 3 we provide an example of the applied virtues of ∀ ∃ \forall \exists -correct subsemirings of the isols.