Combinatorial Isols and the Arithmetic of Dekker Semirings

Abstract

In his long and illuminating paper [1] Joe Barback defined and showed to be non-vacuous a class of infinite regressive isols he has termed "completely torre" (CT) isols. These particular isols all enjoy a property that Barback has since labelled combinatoriality. In [2], he provides a list of properties characterizing the combinatorial isols. In Section 2 of our paper, we extend this list of characterizations to include the fact that an infinite regressive isol X is combinatorial if and only if its associated Dekker semiring D(X) satisfies all those Π2 sentences of the language LN for isol theory that are true in the set ω of natural numbers. (Moreover, with X combinatorial, the interpretations in D(X) of the various function and relation symbols of LN via the "lifting" to D(X) of their Σ1 definitions in ω coincide with their interpretations via isolic extension.) We also note in Section 2 that Π2(L)-correctness, for semirings D(X), cannot be improved to Π3(L)-correctness, no matter how many additional properties we succeed in attaching to a combinatorial isol; there is a fixed ∀ ∃ ∀ (L) sentence that blocks such extension. (Here L is the usual basic first-order language for arithmetic.) In Section 3, we provide a proof of the existence of combinatorial isols that does not involve verification of the extremely strong properties that characterize Barback's CT isols.Mathematics Subject Classification: 03D50, 03F30.