Our paper contains three theorems on regressive isols that are hereditarily odd-even. Two are characterizations of hereditarily odd-even isols in terms of a parity property of the isol and a property on the comparability of summands of the isol. In the third theorem, we show that if a regressive isol has a special comparability of summands property, then it has a predecessor that is hereditarily odd-even.
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.
In this paper we introduce a collection of isols having some interesting properties. Imagine a collection W of regressive isols with the following features: (1) u , v E W implies that ZL 5 v or v 5 u, ( 2 ) ZL 5 v and v E W imply ZL E W, (3) W contains H = { 0 , 1 , 2 , . . .} and some infinite isols, and (4) u E W , u infinite, and u + v regressive imply u + v E W.That such a collection W exists is proved in our paper. It has many nice features. It also satisfies (5) u , v E W , u 5 v and u infinite imply v 5 g a ( u ) for some recursive combinatorial function g, and (6) each u E W is hereditarily odd-even and is hereditarily recursively strongly torre. The collection W that we obtain may be characterized in terms of a semiring of isols D ( c ) introduced by J. C. E. Dekker in [5]. We will show that W = D ( c ) , where c is an infinite regressive is01 that is called completely torre.Mathematics Subject Classification: 03D50.
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