Some properties of ∀∃ models in the isols

McLaughlin, T. G.

doi:10.1090/S0002-9939-1986-0840636-X

Some properties of $\forall \exists$ models in the isols
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by T. G. McLaughlin

Proc. Amer. Math. Soc. 97 (1986), 495-502

DOI: https://doi.org/10.1090/S0002-9939-1986-0840636-X

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Abstract:

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 ${\mathbf {R}}$ of isols that are models of $\forall \exists$ arithmetic, and we show that all such ${\mathbf {R}}$ are in fact models of the $\omega$-true $\forall \exists$ sentences of isol theory. We solve one of the chief problems left open in [8], and in $\S 3$ we provide an example of the applied virtues of $\forall \exists$-correct subsemirings of the isols.

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