A fine structure in the theory of isols. (English) Zbl 0903.03027
The author’s abstract: We introduce a collection of isols having some interesting properties. Imagine a collection \(W\) of regressive isols with the following features: (1) \(u,v\in W\) implies that \(u\leq v\) or \(v\leq u\). (2) \(u\leq v\) and \(v\in W\) imply \(u\in W\). (3) \(W\) contains \(N= \{0,1,2,\dots\}\) and some infinite isols, and (4) \(u\in W\), \(u\) infinite, and \(u+v\) regressive imply \(u+ v\in W\). That such a collection \(W\) exists is proved in our paper. It has many nice features. It also satisfies (5) \(u,v\in W\), \(u\leq v\) and \(u\) infinite imply \(v\leq g_\wedge(u)\) for some recursive combinatorial function \(g\), and (6) each \(u\in W\) 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 \({\mathcal D}(c)\) introduced by J. C. E. Dekker. We show that \(W={\mathcal D}(c)\), where \(c\) is an infinite regressive isol that is called completely torre.
Reviewer: R.Sh.Omanadze (Tbilisi)
MSC:
| 03D50 | Recursive equivalence types of sets and structures, isols |
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