Patterns of resemblance of order 2
This article continues the project of the author into developing ordinal notation systems based on \(\Sigma_n\)-elementary substructures. The basic idea is to consider finite substructures of the structures \((\text{ORD},0,+,\leq,\leq_1 ,\dots, \leq_n)\), where \(\leq_i\) is interpreted as \[ \alpha \leq_i \beta \iff (\alpha,0,+,\leq,\leq_1, \leq_n) \prec_{\Sigma_i} (\alpha,0,+,\leq,\leq_1 ,\dots, \leq_n). \] This idea is then varied. Here the operations are considered as partial operations. Let \(B=(A,0,+,\leq,\leq_1 ,\dots, \leq_n)\) be a minimal final substructure w.r.t. pointwise ordering. This structure can be described finally. Then \(B\) together with an \(a \in A\) can be used in order to denote the ordinal by which \(a\) is interpreted. This way, one obtains notations for ordinals. The set of ordinals which can be denoted this way forms the core of the structure. The goal of this project is to derive computable well-orderings based on this idea and to find an ultrafine structure for large-cardinal axioms based on such embeddings. Previous work as part of this project has influenced the development of ordinal notation systems of strength \(\Pi^1_2\)-CA and beyond by Michael Rathjen. This work can be seen as well as a project on developing strong natural ordinal notation systems, which originate from purely structural set-theoretic investigations rather than from defining an ordinal notation systems based on functions (which don't have a direct justification of being natural and therefore can always be varied in order to obtain quite different notation systems) -- however, the author doesn't make any claims in this direction, he does not even use the term ``natural well-ordering''. In this article the author studies patterns of resemblance of order 2, i.e.~\(n\) as above is \(2\). He computes the core of these patterns of resemblance and shows that it is a computable structure. The structure of the article is as follows: The paper presents first (Section~1) some background material and introduces (Section~2) a generalisation of the notions of isominimality and core, given by the author before for general linearly ordered structures. In Section~3, Ehrenfeucht and Mostowski structures (EM) are introduced, which restrict the linearly ordered structures used. All structures used will follow these restrictions. In Section~4, arithmetic structures are introduced. In Section~5, the variant used here for patterns of resemblance of order 2 is introduced (Definition~5.6). Rules for extending patterns of resemblance are studied in Section~6. An extension of a pattern is valid if coverings of the original patterns can be extended in a unique way. In Section~7, the notion of continuity between a pattern and its extension is studied. Arithmetic and transcendental extensions of patters are studied in Section~8. In Section~9, the concept of downward reflection is studied, which corresponds to the reflection of a structure down into a given interval. This resembles very much the reflection principles used in Michael Rathjen's articles on \(\Pi^1_2\)-CA. In Section~10, upward reflection is studied and in Section~11, arithmetic liftings are investigated. In Section~12, the lifting of patterns is presented. Sections~10--12 provide various ways of extending patterns. In Sections~13--14, rules for extending patterns are given which are sufficient for generating the core of patterns of resemblance of order 2. In Section~15 it is shown that the core is computable. The article ends with some concluding remarks.
scientific article; zbMATH DE number 5542408
scientific article