Patterns of resemblance: Difference between revisions

The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS.

A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref>T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref>https://arxiv.org/pdf/1710.01870.pdf</ref> For all systems analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref>