Hereditarily Odd–Even and combinatorial isols

Abstract

In this paper we study some of the arithmetic structure that is found in a special kind of semi-ring in the isols. These are the semi-rings [D(Y ), +, ·] that were introduced by J.C.E. Dekker, and that were later shown by E. Ellentuck to model the true universal recursive statements of arithmetic when Y is a regressive isol and is hyper-torre (= hereditarily odd-even = HOE). When Y is regressive and HOE, we further reflect on the structure of D(Y ). In addition, a new variety of regressive isol is introduced, called combinatorial. When Y is such an isol, then it is also HOE, and more, and the arithmetic of D(Y ) is shown to have a richer structure.

On hyper‐torre isols

“…Since the predecessor of a regressive isol that is HOE is also HOE, it then follows from the definition of D(Y ) that if Y is regressive and HOE, then all of the members of D(Y ) are HOE. The contents of the following theorem are proved in [2], and a proof will be omitted here. The result provides a characterization of the HOE property for regressive isols in terms of the extension to the isols of recursive elementary functions.…”

Combinatorial Isols and the Arithmetic of Dekker Semirings

“…(1) has already been proved in [8] for the case of function symbols, and from there the extension to include relation symbols is, as we shall see, routine. Given (1), (2) follows from [8,Theorem 2] (which says that (Γ, + , • ) satisfies the ω-true universalexistential sentences of L N but with function and relation symbols interpreted by their isolic extensions). Thus we do not have far to go from [8] to establish (1) and (2).…”

“…Given (1), (2) follows from [8,Theorem 2] (which says that (Γ, + , • ) satisfies the ω-true universalexistential sentences of L N but with function and relation symbols interpreted by their isolic extensions). Thus we do not have far to go from [8] to establish (1) and (2). Claim (3) will be obtained directly from (2), by induction on the number of bounded quantifiers and a trivial lemma that is needed to push the induction.…”

“…Thus we do not have far to go from [8] to establish (1) and (2). Claim (3) will be obtained directly from (2), by induction on the number of bounded quantifiers and a trivial lemma that is needed to push the induction. Some compact notation will be helpful.…”

“…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.…”