Patterns of resemblance: Difference between revisions
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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 version 4]]. 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 name="ElementaryPatterns">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 name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref name="OrdinalArithmeticSigmaOne">T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997)
A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" /><sup>p.23</sup><ref name="PureSigma2Beyond" /><sup>p.6</sup> (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this)
==Reflection criterion==
Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns.
==Stability==
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