Patterns of resemblance: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "The patterns of resemblance (PoR) are a system of ordinal-notations introduced by TJ Carlson. It is superficially similar to stability, yet is a notation for recursive rather than nonrecursive ordinals, and uses elementary substructures between ordinals themselves, instead of between ranks of the constructible universe. It uses a structure also found in BMS known as respecting forests, and was originally believed to have the same limit as BMS.") |
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The patterns of resemblance (PoR) are a system of ordinal-notations introduced by
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). Accessed 29 August 2023.</ref><sup>implicit in section 3</sup><ref name="ElementaryPatterns" /><sup>corollary 6.12</sup><ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007208001760 Patterns of resemblance of order two]", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref>
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. (I think <ref name="OrdinalArithmeticSigmaOne" /> is a citation)
==Stability==
It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" /><sup>p.20</sup> so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS.
==Citations==
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