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 “complete y torre” isols. These particular isols a enjoy a property that Barback has since labelled combinatoriality. In [2], he provides a list of properties characterizing the combinatoria 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 satisfies all those Π2 sentences of the anguage LN for isol theory that are true in the set ω of natural numbers. of the various function and relation symbols of LN via the “lifting ” to D of their Σ1 definitions in ω coincide with their interpretations via isolic extension.) We also note in Section 2 that Π2-correctness, for semirings D, cannot be improved to Π 3-correctness, no matter how many additional properties we succeed in attaching to a combinatoria isol; there is a fixed equation image sentence that blocks such extension. 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